Functions (Edexcel A Level Maths: Pure): Exam Questions

The functions and are defined by

State the range of

Find

Find

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the graph with equation

Solve

The point lies on the curve with equation

Find the point to which is mapped, when the curve with equation is transformed to the curve with equation

(i) Sketch the graph of .

(ii) Explain why for all real values of .

The function is defined as

Sketch the curve , labelling the coordinates of any points where the curve meets the coordinate axes.

Find and label the coordinates of the turning point on your sketch.

Find the range of .

Find the largest possible domains of the following functions:

(i)

(ii)

(iii)

On the same axes, sketch the graphs of  and  where

Label the points at which the graphs meet the coordinate axes.

Solve the equation 

The function is defined as

 Show that can be written in the form

where and are constants to be found.

Explain why the inverse of does not exist.

The domain of is changed to .

 Find and state its domain and range.

State whether the following mappings are one-to-one or many-to-one:

(i)      

(ii)

(iii)     

(iv)

The functions and are defined as follows:

Find and simplify

(i)

(ii)

Explain why and state the domain and range of .

The functions and are defined as follows:

Sketch the graph of and label the coordinates of any points where the graph meets the coordinate axes.

Find the number of solutions to the equation

in the cases when:

(i)

(ii)

Solve

The functions and are defined by:

Find the range of .

Find expressions for

(i) 

(ii) 

Give your answers in the form where , and are constants to be found.

Solve the equation

The graph of is shown below.

The dotted line has the equation

On the diagram, sketch the graph of .

On the same axes, sketch the graphs of   and  where

Label the coordinates of the points at which the graphs meet the coordinate axes.

Solve the equation

Which of the solutions to is also a solution to ?

Find .

State the range of .

Find , stating its domain.

The function is defined by

Find

Show that

where and are constants to be found.

The function is defined by

State the range of

Find the range of

Figure 4 shows a sketch of the graph of , where

Find the value of .

Find all the values of for which

The function is defined by

Explain why has an inverse but does not.

Solve

Figure 2 shows a sketch of the graph with equation

The vertex of the graph is at the point , shown in Figure 2.

Find the coordinates of .

Solve the equation

A line has equation , where is a constant.

Given that intersects at least once,

find the range of possible values of , writing your answer in set notation.

Write in the form , where and are integers to be found.

Sketch the curve with equation showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Find the range of the function

Solve the equation , giving your answers in exact form.

The functions and are defined by

Find and simplify expressions for

(i)

(ii)

Explain why  and state the domain and range of .

Describe the transformation that would map the graph of on to the graph of .

The functions and are defined by

Sketch the graph of .

Label clearly the coordinates of any points where the graph meets the coordinate axes.

Find the number of solutions to the equation

in the cases when:

(i)

(ii)

Solve the equation

The functions and are defined by

Find the range of .

Leaving your answers as single algebraic fractions, find expressions for

(i)  

(ii)

Use algebra to solve the equation

On the same axes, sketch the graphs of  and  where

Label the points at which the graphs meet the coordinate axes.

Solve the equation

Which (if any) of the solutions to  are not solutions to ?

A function is defined by

where .

The graph of is shown below, where

• is the local maximum point

• is the -intercept

Find the values of  and

Solve

Solve the equation 

giving your answers in exact form.

The function is defined by

Find .

Show that where and are integers to be found.

Figure 4 shows a sketch of the graph with equation

where is a positive constant.

Sketch the graph with equation where

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of , the set of values of for which

giving your answer in set notation.

The function is defined by

State the range of

Find

The function is defined by

Find

Find the exact value of the constant for which

The function is defined by

Find the range of .

You must show your working clearly.

The domain of is changed to 

Find the range of .

Find a different domain of that has the same range as in part (b).

The graph of is shown in the diagram below. The dotted line has equation .

You are given that

• has rotational symmetry about the origin

• For , the vertices of the graph have coordinates , and

• does not continue for

(i) Find the domain and range of the function .

(ii) On the diagram above, sketch the reflection of in the line  and explain why this cannot be the graph of 

(i) Find the largest restricted domain of  that includes the value such that  exists.

(ii) Assuming the domain of in part (b)(ii) is used, find the domain and range of the function .

The function is defined by

Determine whether or not the inverse of  exists.

The domain is changed such that

• the inverse of exists

• the graph of lies in the first quadrant only

• the domain of is as large as possible

Find the new domain and range of .

The domain of is changed to .

Find an expression for .

State also the domain and range of .

The functions and are defined by

Find and simplify expressions for

(i) 

(ii) 

Write down and state its domain and range.

Describe the transformation that would map the graph of on to the graph of .

Find the coordinates of the point of intersection between the curves  and .