A LevelMaths: PureOCRExam Questions2. Algebra & Functions2.8 Functions (A Level only)

Functions (A Level only) (OCR A Level Maths: Pure)

Exam Questions

3 hours33 questions
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1
Sme Calculator4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

 

(i)
straight f colon space x rightwards arrow from bar 4 x minus 2
(ii)
straight f colon space x rightwards arrow from bar x squared
(
iii)
straight f colon space x rightwards arrow from bar x over 4
(iv)
straight f colon space x rightwards arrow from bar square root of x

 

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2
Sme Calculator3 marks

State the largest possible domains for the following functions.

(i)
straight f colon space x rightwards arrow from bar square root of x
(ii)
straight f colon space x rightwards arrow from bar ln left parenthesis x minus 2 right parenthesis
(iii)
straight f colon space x rightwards arrow from bar arcsin space x

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3
Sme Calculator3 marks

State the range for the following functions based on the given domains.

(i)
straight f colon space x rightwards arrow from bar e to the power of x               x element of straight real numbers
(ii)
straight f colon space x rightwards arrow from bar x squared space plus space 1        x element of straight real numbers
(iii)
straight f colon space x rightwards arrow from bar 1 over x               x element of straight real numbers

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4a
Sme Calculator3 marks

The function f(x) is defined as

straight f left parenthesis x right parenthesis space equals space x to the power of 2 space end exponent minus space 8 x space minus space 20 space space space space space space space space space space x element of straight real numbers

Sketch the graph of y = f(x), giving the coordinates of any points where the graph intersects the coordinate axes.

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4b
Sme Calculator2 marks

The minimum point on the graph of y = f(x) has x -coordinate 4.
Find the range of f(x) .

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5
Sme Calculator3 marks

The function f(x)  is defined as

straight f left parenthesis x right parenthesis space equals space x squared space space minus space space 9 space space space space space space space space space space x space greater or equal than space 3   

(i)
Work out the range of f(x).
(ii)
If the domain of f(x) is changed to x space greater or equal than space 0 comma what would be the new range of f(x)?

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6a
Sme Calculator4 marks

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

 straight f left parenthesis x right parenthesis space equals space 3 x space plus space 5 space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis space equals space minus 2 x space space space space space space space space space space space space space space space x element of straight real numbers space

Find

(i)
fg(x)   

(ii)
gf(x) 

 

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6b
Sme Calculator2 marks

Solve the equation f(x) = g(x) .

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7a
Sme Calculator3 marks

The function f(x) is defined by

 straight f left parenthesis x right parenthesis space equals space 3 x squared space plus space 1 space space space space space space space space space space straight x element of straight real numbers

Find the inverse of f(x) , f−1(x).

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7b
Sme Calculator2 marks

Find the domain and range for f-−1(x).

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8a
Sme Calculator3 marks

Solve the equation vertical line 6 minus 2 x vertical line space equals space 4.

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8b
Sme Calculator4 marks

On the same diagram, sketch the graphs of y space equals space vertical line 6 minus 2 x vertical line and y space equals space 4.
Label the coordinates of the points where the two graphs intersect each other and the coordinate axes.

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8c
Sme Calculator3 marks

Consider the graphs of y space equals vertical line 6 minus 2 x vertical line and y equals k, where k is a constant.
For which values of  k...

(i)
... will the two graphs have no points of intersection?

(ii)
... will the two graphs have one point of intersection?

(iii)
... will the two graphs have two points of intersection?

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1
Sme Calculator4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i)
straight f colon x rightwards arrow from bar x squared
(ii)
straight f colon x rightwards arrow from bar 3 x plus 1
(iii)
straight f colon x rightwards arrow from bar open parentheses x plus 1 close parentheses cubed
(iv)
straight f colon x rightwards arrow from bar plus-or-minus square root of x

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2a
Sme Calculator3 marks

The function straight f left parenthesis x right parenthesis is defined as

straight f left parenthesis x right parenthesis space equals space x squared space plus space 2 x space long dash space 3 space space space space space space space space space space space space x element of straight real numbers

Sketch the graph of y space equals space straight f left parenthesis x right parenthesis, giving the coordinates of any points where the graph intercepts the coordinate axes and the coordinates of the turning point.

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2b
Sme Calculator1 mark

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

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3a
Sme Calculator2 marks

The functionspace straight f left parenthesis x right parenthesis spaceis defined as

straight f left parenthesis x right parenthesis space equals space x squared space long dash space 4 space space space space space space space space space space x greater or equal than 0

Work out the range of straight f open parentheses x close parentheses.

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3b
Sme Calculator1 mark

If the domain of straight f left parenthesis x right parenthesis is changed to x space less or equal than space 0, what is the range of straight f left parenthesis x right parenthesis?

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4a
Sme Calculator1 mark

The functions straight f left parenthesis x right parenthesis and space straight g left parenthesis x right parenthesisare 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 space space space space space space space space space space space x element of straight real numbers g left parenthesis x right parenthesis space equals space 4 x space long dash space 3 space 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

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

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4b
Sme Calculator4 marks
Find 
(i)
straight f g left parenthesis x right parenthesis
(ii)
g straight f left parenthesis x right parenthesis

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4c
Sme Calculator2 marks

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

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5a
Sme Calculator3 marks

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

q5a-1-2-8-medium-aqa-a-level-maths

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

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5b
Sme Calculator2 marks

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

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6a
Sme Calculator3 marks

On the same axes, sketch the graphs of