Functions

Express the inverse function in the form 

Find 
Simplify your answer.

f is a function such that

Find

 is a function such that

Find

Give your answer as simply as possible.

is the function 

Find

Express the inverse function  in the form =

is the function 

Find

The functions  and  are defined as

Work out

Work out  

Work out the value of .

Express the inverse function  in the form  = ...

f is the function such that  f(x) = 2x - 5
g is the function such that g(x) = x² - 10

Find f (4)

Find fg(-4)

Express the inverse function f^-1 in the form f^-1 (x) = ...

Solve gf (x) = -1

The functions and are such that

Find the value of

Find 

Show that 

The function f is defined as f() = 

State which value of  must be excluded from any domain of f.

The function is defined as = 5 +

Given that = 7, find the value of .

Calculate (1)

Find 
Simplify your answer.

The function is defined as

State the value of that cannot be included in the domain of h

Express the inverse function h^–1 in the form

.................... 

The function  is such that for all values of .

Find

State the range of the function

The function  is such that 

Work out

Circle the expression for

A function is represented by the following function machine.

A number is input into the machine.
The output is used as a new input.
The second output is 11.

Work out the number that was the first input.

A number is input into the machine.
The output given is the same number.

Work out the number.

and 

Circle the expression for

The functions  and  are such that

where  and  are constants.

Find the value of  and the value of .

Find f(10)

State which values of  must be excluded from a domain of f

Find f

One of the solutions of , where  is a number, is 

Find the other solutions
Give your answers correct to 1 decimal place

For which values of  does the equation  have only one solution?

Find an estimate for the gradient of the curve at the point where  = −2.5

State which value of  must be excluded from any domain of 

The function is defined as

       with domain

Express the inverse function g⁻¹ in the form

................................................... 

State the domain of 

The functions and are such that

The function is such that

The domain of is

Express the inverse function in the form

The functions and are defined as

Find

Express the inverse function in the form 

...............................

The functions and are such that

The function  is such that 

Express the inverse function  in the form  = .........

= ................

The function is such that       for

Express  in the form  where  and  are constants.

Using your answer to part (a), find the range of values of  for which  is positive.

 is the function with domain such that 

Write down the range of 

Express the inverse function  in the form 

The function f is defined as for  and where  is a positive number.

Find the value of  for which 

The function  is defined as

Given that 
find an expression for  in terms of .

Solve 

Work out the value of  

  and     

Simplify 

  for all values of

On the grid, draw the graph of the inverse function   for  

For all values of

On the grid, draw the graph of the composite function    for 

Here are two function machines, A and B.

Both machines have the same input.

Work out the range of input values for which

                                   the output of A is less than the output of B.

and are constants.

Work out the value of .