DP IB Maths: AA HL

Topic Questions

2.3 Functions Toolkit

1a
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3 marks

The functions f  and g  are defined such that  f left parenthesis x right parenthesis equals 2 x squared minus 4 x   and   g left parenthesis x right parenthesis equals fraction numerator 5 x space plus space 12 over denominator 2 end fraction.

Find left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis comma , giving your answer in the form  left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals m left parenthesis x minus h right parenthesis squared plus k  where m , h  and k  are constants to be found.

1b
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1 mark

Hence, or otherwise, find the coordinates of the vertex of the graph of  y equals left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis. .

1c
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3 marks

Find left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis ,  giving your answer in the form  left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals a x squared plus b x plus c   where a ,b  and c  are constants to be found.

1d
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1 mark

Hence, or otherwise, find the coordinates of the y -intercept of the graph of  y equals left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis.

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2a
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1 mark

Let  f left parenthesis x right parenthesis equals fraction numerator 5 minus x squared over denominator 3 end fraction  and g left parenthesis x right parenthesis equals 4 minus 3 over x  ,  where each function has the largest possible valid domain.

Write down the range of f

2b
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2 marks

Write down the domain and range of g.

2c
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3 marks

Find

          (i)      left parenthesis f space ring operator space g right parenthesis left parenthesis x right parenthesis                                

          (ii)      left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis.

2d
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2 marks

Solve the equation left parenthesis f space ring operator space g right parenthesis left parenthesis x right parenthesis equals left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis..

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3a
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2 marks

The function f is defined by  f left parenthesis x right parenthesis equals cube root of 4 left parenthesis 1 minus x right parenthesis end root ,  for negative 1 less or equal than x less or equal than 17 .

Write down the range of f.

3b
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2 marks

Write down an expression for f to the power of negative 1 end exponent.

3c
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2 marks

Write down the domain and range of f to the power of negative 1 end exponent

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4a
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4 marks

The perimeter, P, and area, A, of a given square can be expressed by straight P equals 4 x and straight A equals x to the power of italic 2 respectively, where x is the length of the side of the square.

Write down an expression for:

(i)

P in terms of A,

(ii)
A in terms of P, A left parenthesis P right parenthesis.
4b
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2 marks

straight P to the power of negative 1 end exponent left parenthesis 40 right parenthesis equals straight A left parenthesis k right parenthesis

Find the value of k spaceand straight A open parentheses straight k close parentheses.

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5a
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1 mark

The values of two functions, f and g, for certain values of  are given in the following table:

x negative 2 0 3
f left parenthesis x right parenthesis negative 12 negative 4 8
 g left parenthesis x right parenthesis 0 negative 12 30

Find the value of f to the power of negative 1 end exponent open parentheses 8 close parentheses.

5b
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2 marks

Find the value of left parenthesis f ring operator g right parenthesis left parenthesis negative 2 right parenthesis.

5c
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2 marks

Given that f left parenthesis x right parenthesis is a linear function, find f left parenthesis x right parenthesis.

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6a
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3 marks

Let  f left parenthesis x right parenthesis equals square root of x minus 14 end root comma  for x greater or equal than 14.

Find f to the power of negative 1 end exponent left parenthesis 2 right parenthesis.

6b
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3 marks

Let g be a function such that g to the power of negative 1 end exponent exists for all real numbers.

Given that g left parenthesis 14 right parenthesis equals 3, find left parenthesis f ring operator g to the power of negative 1 end exponent right parenthesis right parenthesis left parenthesis 3 right parenthesis.

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7a
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4 marks

Let the function f be defined by f left parenthesis x right parenthesis equals square root of 2 x squared minus 16 x plus 41 end root ,  where f has its largest possible valid domain.

Find the domain and range of f.

7b
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2 marks

(i)     Find the value(s) of x for which  f left parenthesis x right parenthesis equals square root of 11 .

(ii)    Use your answer to part (b)(i) to explain why the inverse function f to the power of negative 1 end exponent does not exist.

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8a
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2 marks

Let f left parenthesis x right parenthesis equals x squared minus 9 and g left parenthesis x right parenthesis equals x squared minus 1, both for x greater or equal than 0.

Find

(i)

f to the power of negative 1 end exponent left parenthesis x right parenthesis

(ii)
g to the power of negative 1 end exponent left parenthesis x right parenthesis
8b
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2 marks

Find left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis in the form a x to the power of 4 plus b x squared plus c.

8c
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3 marks

Solve the equation left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals 0.

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9a
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2 marks

Express x squared plus 12 x plus 24 in the form a left parenthesis x plus b right parenthesis squared plus c, where a comma b comma c element of Z.

9b
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3 marks

Given that g left parenthesis x right parenthesis equals x plus 6 and left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals x squared plus 12 x plus 24, find f left parenthesis x right parenthesis.

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10a
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2 marks

Write 2 x squared plus 8 x minus 3  in the form  a left parenthesis x plus h right parenthesis squared plus k.

10b
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1 mark

Explain why the function f  defined by f left parenthesis x right parenthesis equals 2 x squared plus 8 x minus 3 comma   x element of straight real numbers,  does not have an inverse.

10c
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6 marks

The function g defined by   g left parenthesis x right parenthesis equals 2 x squared plus 8 x minus 3 comma   x greater or equal than p  has an inverse.

  (i)      Write down the smallest possible value of .

Given that p  takes its smallest possible value:

   (ii)     Find the domain and range of g to the power of negative 1 end exponent.

   (iii)    Find the inverse function g to the power of negative 1 end exponent .

10d
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3 marks

Solve left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals 21.

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11a
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4 marks

Let f left parenthesis x right parenthesis be an even function and let g left parenthesis x right parenthesis be an odd function.  Both functions are defined for all real values of x.

Prove the following statements:

p left parenthesis x right parenthesis equals f left parenthesis x right parenthesis g left parenthesis x right parenthesis spaceis an odd function.

q left parenthesis x right parenthesis equals left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis spaceis an even function.

11b
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3 marks

Determine whether or not it is possible for the function r spacedefined by 

r left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis

to be even or odd, being sure to state clearly any conditions that apply.

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12
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6 marks

The function f is defined by  f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction ,  where a,b c,  and d are real constants with  c not equal to 0 .

Given that f is a self-inverse function, find the value of a plus d.

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1a
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2 marks

The functions f and g are defined such that f left parenthesis x right parenthesis equals 4 x minus 10 and g left parenthesis x right parenthesis equals fraction numerator x space plus space 8 over denominator 2 end fraction.

Show that left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis equals 2 x minus 1.

1b
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2 marks

Given that left parenthesis g ring operator f right parenthesis left parenthesis a right parenthesis equals 27, find the value of a

1c
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2 marks

Show that left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals 2 x plus 6.

1d
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2 marks

Given that left parenthesis f ring operator g right parenthesis left parenthesis b right parenthesis equals 44, find the value of b

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2a
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1 mark

The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are defined as follows

f left parenthesis x right parenthesis equals x squared space space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis equals 4 x minus 3 space space space space space space space space space space space x element of straight real numbers

Write down the range of f left parenthesis x right parenthesis.

2b
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4 marks

Find    

(i)      left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis           

(ii)      left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis                   

2c
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2 marks

Solve the equation f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis.

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3a
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3 marks

The graph of y equals f left parenthesis x right parenthesis is shown below.

2-8-m-q5-edexcel-al-maths-pure

(i)     Use the graph to write down the domain and range of f left parenthesis x right parenthesis

(ii)    Given that the point (1, 1) lies on the dotted line, write down the equation of the line.

3b
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2 marks

On the diagram above sketch the graph of y equals f to the power of negative 1 end exponent left parenthesis x right parenthesis

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4a
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2 marks

The function f left parenthesis x right parenthesis spaceis defined as

              f left parenthesis x right parenthesis equals fraction numerator x squared plus 1 over denominator x squared end fraction space space space space space space space space space space space space space space space space space space space x element of R comma x not equal to 0

Show that f left parenthesis x right parenthesis spacecan be written in the form

f left parenthesis x right parenthesis equals 1 plus 1 over x squared

4b
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2 marks

Explain why the inverse of f left parenthesis x right parenthesis spacedoes not exist and suggest an adaption to its domain so the inverse does exist.

4c
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4 marks

The domain of f left parenthesis x right parenthesis spaceis changed to x greater than 0.
Find an expression for f to the power of negative 1 end exponent left parenthesis x right parenthesis  and state its domain and range.

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5a
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3 marks

The functions  f left parenthesis x right parenthesis  and  g left parenthesis x right parenthesis  are defined as follows

f left parenthesis x right parenthesis equals 1 half space open parentheses 4 x minus 3 close parentheses space space space space space space space space x element of R

g left parenthesis x right parenthesis equals 0.5 x plus 0.75 space space space space space space space space space x element of R

Find

(i)

left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis

(ii)
left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis
5b
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3 marks

Write down f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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6a
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1 mark

A function is defined by f left parenthesis x right parenthesis equals 54 x minus 13 comma space space space space minus 2 less than x less than 20.

Find the value of f left parenthesis begin inline style 5 over 2 end style right parenthesis.

6b
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2 marks

Write down the range of f left parenthesis x right parenthesis.

6c
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2 marks

Find the inverse function f to the power of negative 1 end exponent left parenthesis x right parenthesis.

6d
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1 mark

Write down the range of the inverse function.

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7a
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2 marks

Consider the function f left parenthesis x right parenthesis equals negative 6 x minus 3. The domain of f left parenthesis x right parenthesis space space space space minus 5 less or equal than x less or equal than 3. 

Find

(i)

f left parenthesis 2 right parenthesis

(ii)
x space when space f left parenthesis x right parenthesis equals 15
7b
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3 marks

Find the range of f left parenthesis x right parenthesis.

7c
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3 marks

Write down an expression for f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain.

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8a
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3 marks

Consider the function g left parenthesis x right parenthesis equals space square root of left parenthesis 4 minus x right parenthesis end root.

Sketch the graph of the function g left parenthesis x right parenthesis, labelling the x spaceand y

ib8a-ai-sl-2-3-ib-maths-medium

8b
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2 marks

Find

    (i)     straight g left parenthesis negative 5 right parenthesis

    (ii)    x  when  straight g left parenthesis x right parenthesis equals space 1 half .

8c
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2 marks

Find

          (i)      the maximum possible domain of the function straight g left parenthesis x right parenthesis

(ii)
      the range of the function that corresponds to the domain found in part (c) (i).

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9a
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2 marks

The functions f and g are defined for x element of straight real numbers by f left parenthesis x right parenthesis equals 3 x squared plus 10 x plus 7 and   g left parenthesis x right parenthesis equals x plus d comma space where d element of R.

Find the range of f.

9b
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4 marks

Given that left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis is always positive for all x determine the set of possible values for d

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10a
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2 marks

Let  f open parentheses x close parentheses equals fraction numerator 2 x minus 5 over denominator x plus 8 end fraction, where x not equal to a comma space x element of R., .

Write down

(i)

the value of a

(ii)
the range of f.
10b
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1 mark

For the graph of f, find the equations of all the asymptotes.

10c
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2 marks

Find f to the power of negative 1 end exponent open parentheses x close parentheses.

10d
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2 marks

For the graph of f to the power of negative 1 end exponent, find the equation of

(i)

the horizontal asymptote

(ii)
the vertical asymptote.

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11
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5 marks

Determine, for each of the following functions, whether they are even, odd or neither:

(i)

f left parenthesis x right parenthesis equals 1 over x squared plus 2

(ii)

g left parenthesis x right parenthesis equals x cubed minus 3 x

(iii)
h left parenthesis x right parenthesis equals x squared plus 2 x minus 5.

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12
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5 marks

Prove that the sum of two odd functions is also an odd function.

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13a
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2 marks

Let f left parenthesis x right parenthesis equals pi squared over x , where x not equal to 0 comma x element of R .

Show that f left parenthesis x right parenthesis spaceis a self-inverse function.

13b
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1 mark

Let g left parenthesis x right parenthesis equals fraction numerator negative x minus 2 over denominator 5 x plus 1 end fraction, where x not equal to p comma x element of R.

Find the value of p.

13c
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3 marks

Show that g left parenthesis x right parenthesis is a self-inverse function.

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14a
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2 marks

Consider the function f defined by f left parenthesis x right parenthesis equals 2 x cubed plus 3 x squared minus 36 x plus 7 comma x element of R    .

Sketch the graph of f. Clearly label the points where the graph intersects the axes, along with any points that are local maxima or minima.

14b
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3 marks

Let the function g be defined by g left parenthesis x right parenthesis equals 2 x cubed plus 3 x squared minus 36 x plus 7 comma x less or equal than p  .

Given g spacethat has an inverse:

(i)

Find the largest possible value of p

(ii)

Find the domain of for the value of g to the power of negative 1 end exponent  identified in part (b)(i)

(iii)
Find the value of g to the power of negative 1 end exponent open parentheses 0 close parentheses .
14c
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3 marks

Given that h space has an inverse:

(i)

Find the smallest possible value of q

(ii)

Find the domain of h to the power of negative 1 end exponent  for the value of q   identified in part (c)(i)

(iii)
Find the value of h to the power of negative 1 end exponent open parentheses 0 close parentheses .

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1a
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3 marks

The functions f  and g  are defined such that  f left parenthesis x right parenthesis equals 9 x minus 3 x squared minus 3   and g left parenthesis x right parenthesis equals negative fraction numerator 66 plus 2 x over denominator 3 end fraction  , both for  x element of straight real numbers .

Find left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis , giving your answer in the form left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis equals a left parenthesis x minus p right parenthesis left parenthesis x minus q right parenthesis. .

1b
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1 mark

Hence, or otherwise, find the x -intercepts of the graph of y equals left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis..

1c
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6 marks

Let h left parenthesis x right parenthesis equals 1 minus 2 x.

Find the distance between the y-intercept of the graph of  space y equals space left parenthesis f ring operator h right parenthesis left parenthesis x right parenthesis  and the positive x-intercept of the graph of  y equals left parenthesis g space ring operator space f right parenthesis left parenthesis x right parenthesis. space spaceYour answer should be given as an exact value.

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2a
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4 marks

Let the function f be such that  f left parenthesis x right parenthesis equals square root of 5 x squared minus 11 x plus 6.05 end root .

Given that the inverse function f to the power of negative 1 end exponent exists, and that the domain of f is as large as possible,

suggest a domain for f spaceand write down the corresponding range.

2b
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2 marks

Based on your answer to part (a), find f to the power of negative 1 end exponent open parentheses square root of 22.05 end root close parentheses.

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3a
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2 marks

Let f left parenthesis x right parenthesis equals square root of negative 3 x squared plus 8 x plus 16 end root .

Write down the coordinates of the x-intercept of the graph of y equals f left parenthesis x right parenthesis.

3b
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6 marks

Given that f has the largest possible valid domain,

find the domain and range of f.

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4a
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2 marks

Let the function  be defined by f left parenthesis x right parenthesis equals open parentheses 2 x squared minus 5 x minus 12 close parentheses to the power of negative 1 half end exponent minus k  ,  where k is a constant and where f has the largest possible valid domain.

Find the domain of f.

4b
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1 mark

Given that  that limit as x rightwards arrow infinity of f left parenthesis x right parenthesis equals negative 7,  find the value of k.

4c
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3 marks

Write down the equations of any vertical and/or horizontal asymptotes on the graph of y equals f left parenthesis x right parenthesis.

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5a
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4 marks

The following diagram shows the graph of  y equals f left parenthesis x right parenthesis,  for a function f that has the domain negative 3 less or equal than x less or equal than 3.  Point A has coordinates  left parenthesis negative 3 comma 2.5 right parenthesis and point B has coordinates left parenthesis 3 comma negative 2.5 right parenthesis.  The x-intercept of the function is left parenthesis 2 comma 0 right parenthesis as shown.

ib5a-ai-sl-2-3-ib-maths-veryhard

f can be written as a piecewise function, where each of the two pieces is a linear function and where the domain of the first function is negative 3 less or equal than x less or equal than 2.

Write down f left parenthesis x right parenthesis as a piecewise function.

5b
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3 marks

Sketch the graph of  y equals f to the power of negative 1 end exponent left parenthesis x right parenthesis on the same grid above.

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6a
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2 marks

Consider the function h defined by h left parenthesis x right parenthesis equals negative 4 x squared plus 24 x plus 8  , x element of straight real numbers .

Rewrite h left parenthesis x right parenthesis in the form a left parenthesis x plus b right parenthesis squared plus c ,  where a comma b comma c element of straight integer numbers.

6b
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3 marks

Given that  f left parenthesis x right parenthesis equals left parenthesis x minus 3 right parenthesis squared and that left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals h left parenthesis x right parenthesis ,  find g left parenthesis x right parenthesis.

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7a
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4 marks

The functions f and g are defined such that  f left parenthesis x right parenthesis equals fraction numerator 3 minus 2 x over denominator 5 end fraction  and  g left parenthesis x right parenthesis equals 4 x minus 7,  both for   x element of straight real numbers  .

Giving your answers in the form y equals m x plus c,  find

(i)

left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis

(ii)
left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis
7b
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2 marks

Describe a single transformation that would map the graph of y equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis  onto the graph of y equals left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis .

7c
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3 marks

Given that open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis p right parenthesis equals 2,  find the value of p.

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8a
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2 marks

Let the functions f and g be defined by f left parenthesis x right parenthesis equals 9 over 4 x squared minus 1   and  g left parenthesis x right parenthesis equals x squared minus 2,  both for x greater or equal than 0  .

Find

(i)

f to the power of negative 1 end exponent left parenthesis x right parenthesis

(ii)
g to the power of negative 1 end exponent left parenthesis x right parenthesis
8b
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2 marks

Find left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis in the form a x to the power of 4 plus b x squared plus c.

8c
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3 marks

Solve the equation left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals 0.

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9a
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2 marks

A rectangle has length l equals 4 x and width w equals x.

Find an expression for

(i)

the perimeter of the rectangle, P, in terms of x.

(ii)
the area of the rectangle, A, in terms of x.
9b
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2 marks

Show that P left parenthesis A right parenthesis equals 5 square root of A.

9c
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3 marks

The graph of the function P, for 0 less or equal than A less or equal than 4 , is shown below.

ib8c-ai-sl-2-3-ib-maths-veryhard

On the grid above, draw the graph of the inverse function P to the power of negative 1 end exponent

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10a
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5 marks

Consider the function f defined by  f left parenthesis x right parenthesis equals x squared minus 6 x plus 10 comma   x less or equal than p ,  where p is the largest value such that  has an inverse.

  (i)     Find the value of p .

  (ii)    On the same set of axes, sketch the graphs of f and f to the power of negative 1 end exponent.

  (iii)    Write down the domain and range of f to the power of negative 1 end exponent.

10b
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3 marks

Find the inverse function f to the power of negative 1 end exponent .

10c
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4 marks

Let the function g  be defined by  g left parenthesis x right parenthesis equals x squared minus 6 x plus 10 comma   x element of straight real numbers .

  (i)     Solve left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals 2 .

  (ii)    Solve left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals 2..

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11a
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3 marks

Consider the function  f left parenthesis x right parenthesis equals a x to the power of 4 plus b x cubed plus c x squared plus d x plus e where  a comma b comma c comma d comma e element of straight real numbers .

Show that:

(i)

if f space is even then b equals d equals 0.

(ii)
if f space is odd then a equals c equals e equals 0.
11b
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5 marks

Consider the function g  defined by   g left parenthesis x right parenthesis equals left parenthesis 3 x plus p right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis q x plus 1 right parenthesis left parenthesis 2 x plus 3 right parenthesis  where p  and q  are real constants.

Find the possible values of p  and q in the case where g is an even function.

11c
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4 marks

Use proof by contradiction to show that g  can never be an odd function.

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12a
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4 marks

Consider the function f defined by f left parenthesis x right parenthesis equals fraction numerator 2 x minus 5 over denominator 3 x plus k end fraction comma   x element of R comma   x not equal to negative k over 3.

In the case where f  is self-inverse, find the value of k.

12b
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5 marks

In the case when the graphs of f and f to the power of negative 1 end exponent intersect at exactly one point, find the possible values of k.

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13a
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1 mark

A part of the graph of the function f left parenthesis x right parenthesis equals 2 x cubed minus 3 x squared minus 12 x plus 8 comma   x element of straight real numbers is shown below.  

ib13a-ai-sl-2-3-ib-maths-veryhard

Explain why f  does not have an inverse.

13b
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6 marks

The domain of f is now restricted to   a less or equal than x less or equal than b  where   a less than 0  and  b greater than 0 . a and b are chosen so that f has an inverse and the interval left square bracket a comma b right square bracket is as large as possible. 

Find the domain and range of f to the power of negative 1 end exponent

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