DP IB Maths: AI HL

Topic Questions

2.4 Functions Toolkit

1a
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3 marks

The functions f and g are defined such that space f open parentheses x close parentheses equals 2 x squared minus 4 x  and g open parentheses x close parentheses 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 open parentheses x close parentheses,  giving your answer in the form  left parenthesis g space ring operator space f right parenthesis open parentheses x close parentheses equals m open parentheses x minus h close parentheses 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 open parentheses x close parentheses.

1c
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3 marks

Find left parenthesis f ring operator g right parenthesis open parentheses x close parentheses,  giving your answer in the form space open parentheses space f ring operator g close parentheses open parentheses x close parentheses equals a x squared plus b x plus c where a comma space 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 space y equals left parenthesis f space ring operator space g right parenthesis open parentheses x close parentheses.

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2a
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1 mark

Let space f open parentheses x close parentheses equals fraction numerator 5 minus x squared over denominator 3 end fraction space space and  space g open parentheses x close parentheses equals 4 minus 3 over x space,  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 space f open parentheses x close parentheses equals cube root of 4 left parenthesis 1 minus x right parenthesis space end root,  for space minus 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 forspace f to the power of negative 1 end exponent.

3c
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2 marks

Write down the domain and range ofspace 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 space P equals 4 x spaceand A equals x squared respectively, where x is the length of the side of the square.

Write down an expression for: 

(i)
P in terms of A, P(A)

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

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

Find the value of k and A left parenthesis k right parenthesis.

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5a
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1 mark

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

x

-2

0

3

f(x)

-12

-4

8

g(x)

0

-12

30


Find the value of f
-18.

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 (x) is a linear function, find f (x).

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6a
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3 marks

Let space f open parentheses x close parentheses equals square root of x minus 14 end root,  for x greater or equal than 14.

Find f -1(2).

6b
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3 marks

Let g be a function such that g-1 exists for all real numbers.

Given that g(14) = 3, find open parentheses f ring operator g to the power of negative 1 end exponent close parentheses open parentheses 3 close parentheses .

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7a
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4 marks

Let the function f  be defined byspace space f open parentheses x close parentheses 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 open parentheses x close parentheses equals square root of 11 .

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

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8a
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2 marks

Letspace f open parentheses x close parentheses equals x squared minus 9 space andspace g open parentheses x close parentheses 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 open parentheses x close parentheses.

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 open parentheses f ring operator g close parentheses open parentheses x close parentheses equals 0.

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9a
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3 marks
(i)
Show that open parentheses x plus a close parentheses squared plus b equals x squared plus 2 a x plus open parentheses a squared plus b close parentheses

(ii)
Hence show that  x squared plus 12 x plus 24 equals open parentheses x plus 6 close parentheses squared minus 12
9b
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3 marks

Given that gopen parentheses x close parentheses equals x plus 6 and open parentheses f ring operator g close parenthesesopen parentheses x close parentheses equals x squared plus 12 x plus 24, find a possible expression for f open parentheses x close parentheses.

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10a
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3 marks
(i)
Sketch the graph of the function f defined by f open parentheses x close parentheses equals 2 x squared plus 8 x minus 3 comma space x element of straight real numbers, clearly labelling the minimum point with its coordinates.

(ii)
Explain why the function f does not have an inverse.
10b
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3 marks

The function g defined by gopen parentheses x close parentheses 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 p.
Given that p takes its smallest possible value:

(ii)
Find the domain and range of g−1.

(iii)
Find the inverse function g−1.
10c
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3 marks

Solve open parentheses g ring operator f close parentheses open parentheses x close parentheses= 21.

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11
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1 mark

The function f is defined by f open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 4 x minus 1 end fraction.

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

 

It is always true that the graphs of a function and its inverse will be reflections of each other in the line y equals x.

(ii)
Based on the answer to part (i), state what else will be true about the graphs of f and f to the power of negative 1 end exponent.

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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 gleft 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 comma spacefind 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 comma find the value of b.

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2a
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1 mark

The functions f(x) and g(x) are defined as follows

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

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

Write down the range of f(x) .

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(x) = g(x).

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3a
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3 marks

The graph of y = f(x) is shown below.

 2-8-m-q5-edexcel-al-maths-pure

(i)
Use the graph to write down the domain and range of f(x).

(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 = f −1(x).

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4a
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2 marks

The function f open parentheses x close parentheses is defined as

f open parentheses x close parentheses equals fraction numerator x squared plus 1 over denominator x squared end fraction              x element of straight real numbers comma space x ≠ 0 

Show that f open parentheses x close parentheses can be written in the form

f open parentheses x close parentheses equals 1 plus 1 over x squared

4b
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2 marks

Explain why the inverse of f open parentheses x close parentheses does 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 open parentheses x close parentheses is changed to x greater than 0.
Find an expression for f to the power of negative 1 end exponent open parentheses x close parentheses and state its domain and range.

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5a
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3 marks

The functions f open parentheses x close parentheses and g open parentheses x close parentheses are defined as follows

f open parentheses x close parentheses equals 1 half open parentheses 4 x minus 3 close parentheses space space space space space space space space x element of straight real numbers

g open parentheses x close parentheses equals 0.5 x plus 0.75 space space space space space x element of straight real numbers

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 open parentheses x close parentheses and state its domain and range.

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6a
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1 mark

A function is defined by f open parentheses x close parentheses space equals space 54 x minus 13 comma space space minus 2 less than x less than 20.

Find the value of f open parentheses begin inline style 5 over 2 end style close parentheses.

6b
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2 marks

Write down the range of f open parentheses x close parentheses.

6c
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2 marks

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

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 open parentheses x close parentheses space equals space minus 6 x minus 3. The domain of f open parentheses x close parentheses is negative 5 less or equal than x less or equal than 3.

Find

(i)
f open parentheses 2 close parentheses

(ii)
x when f open parentheses x close parentheses space equals space 15.
7b
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3 marks

Find the range of f open parentheses x close parentheses.

7c
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1 mark

Write down the domain of the inverse function.

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8a
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4 marks

Let f open parentheses x close parentheses equals negative fraction numerator 3 over denominator x minus 3 end fraction, for x≠ 3. 

For the graph of f, find:

(i)
the x – intercept

(ii)
the y – intercept

(iii)
the range of f.
8b
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2 marks

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

8c
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2 marks

Given that gopen parentheses x close parentheses equals f open parentheses x plus 3 close parentheses plus 1, find the domain and range of g.

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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 byspace f open parentheses x close parentheses equals 3 x squared plus 10 x plus 7 space and space g open parentheses x close parentheses equals x plus d, where d element of straight real numbers.

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≠ ax element of straight real numbers.

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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11a
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2 marks

Let f open parentheses x close parentheses equals 2 x plus 1for x element of straight real numbers 

Write down an expression for the inverse function f to the power of negative 1 end exponent open parentheses x close parentheses.

11b
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3 marks

Consider another function gopen parentheses x close parentheses equals 1 half open parentheses x minus 1 close parentheses squared plus 3 over 2, for x greater or equal than k where k is an integer to be found.

Given that the graph of g has an inverse, find the value of k.

11c
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4 marks

Sketch the graphs of f and g, for the domain found in part (b), on the same set of axes, along with their inverses.

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12a
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2 marks

Consider the function f defined by f open parentheses x close parentheses equals 2 x cubed plus 3 x squared minus 36 x plus 7 comma space x element of straight real numbers

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.

12b
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3 marks

Let the function g be defined by gopen parentheses x close parentheses equals 2 x cubed plus 3 x squared minus 36 x plus 7 comma space x less or equal than p.

Given that g has an inverse:

(i)
Find the largest possible value of p

(ii)
Find the domain of g1 for the value of p identified in part (b)(i)

(iii)
Find the value of g1open parentheses 0 close parentheses.
12c
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3 marks

Let the function h be defined by  h open parentheses x close parentheses equals 2 x cubed plus 3 x squared minus 36 x plus 7 comma space x greater or equal than q

Given that h 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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13a
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2 marks

A function f is called a self-inverse function if f to the power of negative 1 end exponent open parentheses x close parentheses space equals f open parentheses x close parentheses for all values of x in the domain. 

Let f open parentheses x close parentheses equals straight pi squared over straight x, where x≠ 0, x element of straight real numbers.

Show that f open parentheses x close parentheses is a self-inverse function.

13b
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1 mark

Let gopen parentheses x close parentheses equals fraction numerator negative x minus 2 over denominator 5 x plus 1 end fraction, where x≠ p,x element of straight real numbers

Find the value of p.

13c
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3 marks

Show that gopen parentheses x close parentheses is a self-inverse function.

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1a
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3 marks

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

Findspace left parenthesis g space ring operator space f right parenthesis x, giving your answer in the form  left parenthesis g space ring operator space f right parenthesis x 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 space y equals left parenthesis g space ring operator space f right parenthesis open parentheses x close parentheses.

1c
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3 marks

Let h open parentheses x close parentheses 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 open parentheses x close parentheses space spaceand the positive x-intercept of the graph of space y equals left parenthesis g space ring operator space f right parenthesis open parentheses x close parentheses.  Your 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 open parentheses x close parentheses equals square root of 5 x squared minus 11 x plus 6.05 end root .

It is given that the inverse function f -1 exists, and that the domain of f  is as large as possible,

suggest two possible domains for f and write down the corresponding ranges.

2b
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2 marks

Find what the value of f to the power of negative 1 end exponent open parentheses square root of 22.05 end root close parentheses would be for each of the domains suggested in part (a).

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3a
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3 marks

Letspace f open parentheses x close parentheses equals square root of negative 3 x squared plus 8 x plus 16 space end root.

Write down the coordinates of the y-intercept of the graph of y equals f open parentheses x close parentheses.

3b
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3 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 f be defined by  f open parentheses x close parentheses 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 as x gets large f open parentheses x close parentheses tends towards the value −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 open parentheses x close parentheses.

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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 domainspace minus 3 less or equal than x less or equal than 3.  Point A has coordinates (-3, 2.5) and point B has coordinates (3,-2.5).  The x-intercept of the function is (2, 0) as shown.

D7QW-XUD_q6a-2-2--quadratic-functions-graphs-very-hard-ib-aa-sl-maths

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 open parentheses x close parentheses as a piecewise function.

5b
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3 marks

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

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6a
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3 marks

Consider the function h defined by h open parentheses x close parentheses equals negative 4 x squared plus 24 x plus 8 comma space x element of straight real numbers.

(i)
Show that negative 4 open parentheses x minus 3 close parentheses squared equals negative 4 x squared plus 24 x minus 36.

(ii)
Hence show that h open parentheses x close parentheses equals negative 4 open parentheses x minus 3 close parentheses squared plus 44.
6b
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3 marks

Given that f open parentheses x close parentheses equals open parentheses x minus 3 close parentheses squared and that open parentheses g ring operator f close parentheses open parentheses x close parentheses equals h open parentheses x close parentheses, find a possible expression for gopen parentheses x close parentheses.

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7a
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4 marks

The functions f and g are defined such that space f open parentheses x close parentheses equals fraction numerator 3 minus 2 x over denominator 5 end fraction space spaceand  g open parentheses x close parentheses equals 4 x minus 7,  both for x element of straight real numbers.

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

(i)
open parentheses g ring operator f close parentheses open parentheses x close parentheses

(ii)
open parentheses f ring operator g close parentheses open parentheses x close parentheses.
7b
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2 marks

Describe a single transformation that would map the graph of  y equals open parentheses g ring operator f close parentheses open parentheses x close parentheses  onto the graph of  y equals open parentheses f ring operator g close parentheses open parentheses x close parentheses.

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 open parentheses p close parentheses 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 open parentheses x close parentheses equals 9 over 4 x squared minus 1 and  g open parentheses x close parentheses equals x squared minus 2,  both for  space x greater or equal than 0.

Find

(i)
f -1(x)

(ii)
g -1(x).
8b
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2 marks

Findspace left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis space 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 equationopen parentheses space f ring operator g close parentheses 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 straight A right parenthesis equals 5 square root of straight 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.

q3a-2-2-very-hard-ib-al-

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 open parentheses x close parentheses equals x squared minus 6 x plus 10 comma space x less or equal than p, where p is the largest value such that f 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 gopen parentheses x close parentheses equals x squared minus 6 x plus 10 comma space x element of straight real numbers.

(i)
Solve open parentheses g ring operator f close parentheses open parentheses x close parentheses equals 2

(ii)
Solve open parentheses f ring operator g close parentheses open parentheses x close parentheses equals 2

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11a
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1 mark

A part of the graph of the function f open parentheses x close parentheses equals 2 x cubed minus 3 x squared minus 12 x plus 8 comma space x element of straight real numbers is shown below.  

maths-ai-hl-2-4-q11

Explain why f does not have an inverse.

11b
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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 open square brackets a comma b close square brackets is as large as possible. 

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

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