AQA Level 2 Further Maths · Level 2 Certificate · Chapter 3: Algebra III

Functions, composite and inverse functions, and graphs of linear, quadratic, exponential and piecewise functions

About 90 minutesNo calculator107 marks

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Date

Do Now

These bring back work from chapters 1 and 2 and from GCSE Higher that this chapter uses.

A1
Needed today
Work out the gradient of the line through $(- 4, 5)$ and $(2, - 4)$ .
[2 marks]
Answer

A2
From chapter 2
Write $x^{2} + 14 x + 37$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer
A3
From chapter 2
Make $x$ the subject of $y = \frac{x + 4}{3 x - 2}$ .
[2 marks]
Answer
A4
GCSE Higher
Work out the value of $8^{- \frac{4}{3}}$ .
[2 marks]
Answer
A5
From chapter 1
Expand and simplify $(2 x - 3)^{2} - (x - 3)^{2}$ .
[2 marks]
Answer

Examples, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

$f (x) = \frac{4 x + 1}{x - 3}$ for $x > 3$ .

Work out $f^{- 1} (x)$ and state the domain of $f^{- 1}$ .

Hint Write

y = \frac{4 x + 1}{x - 3}

and make

x

the subject. The domain of

f^{- 1}

is the range of

f

B1
$g (x) = \frac{2 x + 5}{x + 1}$ for $x > - 1$ .
Work out $g^{- 1} (x)$ and state the domain of $g^{- 1}$ .
[4 marks]
Answer $g^{- 1} (x) = \dots\dots\dots$ domain $\dots\dots\dots$

Example 2

The sketch shows the graph of $y = f (x)$ , where $f (x)$ is a quadratic function.

Work out $f (x)$ in the form $a x^{2} + b x + c$ , and the coordinates of the maximum point.

Not drawn accurately

Hint The roots give

f (x) = a (x + 1) (x - 5)

. Use the point on the

y

-axis to find

a

. The maximum point is half way between the roots.

B2
The sketch shows the graph of $y = g (x)$ , where $g (x)$ is a quadratic function.
Work out $g (x)$ in the form $a x^{2} + b x + c$ , and the coordinates of the minimum point.
[4 marks]
Not drawn accurately
Answer $g (x) = \dots\dots\dots (\dots\dots, \dots\dots)$

Example 3

The curve $y = a b^{x}$ passes through the points $(2, 12)$ and $(5, 96)$ .

Work out the values of $a$ and $b$ . Then work out the value of $y$ when $x = - 1$ .

Not drawn accurately

Hint Write two equations,

a b^{2} = 12

and

a b^{5} = 96

. Divide the second by the first so that

a

cancels.

B3
The curve $y = a b^{x}$ passes through the points $(1, 48)$ and $(4, \frac{3}{4})$ .
Work out the values of $a$ and $b$ . Then work out the value of $y$ when $x = 3$ .
[3 marks]
Not drawn accurately
Answer $a = \dots\dots b = \dots\dots y = \dots\dots$

Example 4

$f (x) = {x^{2} + a, b x - 2, 0 ⩽ x < 2 2 ⩽ x ⩽ 5$

The two parts of the graph of $y = f (x)$ join at $x = 2$ , and $f (5) = 18$ .

Work out the values of $a$ and $b$ , and the range of $f$ .

Hint Use

f (5) = 18

first: it is in the second part. Then make the two parts give the same value at

x = 2

B4
Here is a function $f$ .
$f (x) = {a x + 1, x^{2} + b x, - 2 ⩽ x < 1 1 ⩽ x ⩽ 4$
The two parts of the graph of $y = f (x)$ join at $x = 1$ , and $f (4) = 8$ .
Work out the values of $a$ and $b$ , and the range of $f$ .
[4 marks]
Answer $a = \dots b = \dots$ range $\dots\dots$

Practice

Level 2

Each question asks for something different. Show your working.

C1
$f (x) = x^{2} - 5$ . Solve $f (x + 1) = 2 f (x)$ . Give your answers in surd form.
[3 marks]
Answer

C2
$f (x) = 5 - x + 3$ . Write down the largest possible domain of $f$ , and the range of $f$ for that domain.
[2 marks]
Answerdomain $\dots\dots\dots$ range $\dots\dots\dots$
C3
$f (x) = \frac{x}{x - 1}$ for $x \neq = 1$ . Show that $f f (x) = x$ .
[3 marks]
C4
The line $L$ has equation $3 y = 12 - 4 x$ .
(a)
Write down the gradient of $L$ .
[1 mark]
Answer
(b)
Work out the coordinates of the point on $L$ where the $y$ -coordinate is twice the $x$ -coordinate.
[2 marks]
Answer
Total for C4: 3 marks
C5
Work out the equation of the line through $(- 2, 5)$ that is parallel to the line through $(1, - 3)$ and $(5, 7)$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[3 marks]
Answer
C6
The graph of $y = 2 x^{2} + p x + q$ has line of symmetry $x = - 3$ and passes through $(0, 11)$ . Work out the values of $p$ and $q$ , and the least value of $y$ .
[3 marks]
Answer $p = \dots\dots q = \dots\dots$ least value $\dots\dots$
C7
$f (x) = \frac{1}{2} (x - 5)^{2} + 2$ for $x ⩾ 5$ . Work out an expression for $f^{- 1} (x)$ . Hence work out $f^{- 1} (10)$ .
[3 marks]
Answer $f^{- 1} (x) = \dots\dots\dots f^{- 1} (10) = \dots\dots$
C8
The curves $y = 4^{x}$ and $y = 2^{x + 3}$ meet at one point. Work out its coordinates.
[2 marks]
Answer
C9
$f (x) = ⎩ ⎨ ⎧ x + 3, 3, 9 - 3 x, - 3 ⩽ x < 0 0 ⩽ x < 2 2 ⩽ x ⩽ 3$
(a)
Draw the graph of $y = f (x)$ on the grid.
[2 marks]
(b)
Work out the area enclosed by the graph of $y = f (x)$ and the $x$ -axis.
[2 marks]
Answer
Total for C9: 4 marks

Exam-style questions

AQA exam style

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$f (x) = 3 x - 4$ . Kai says, " $f^{- 1} (x) = \frac{1}{3 x - 4}$ ".
Kai is wrong. Explain why, and work out $f^{- 1} (11)$ .
[3 marks]
Answer

D2
$f (x) = 2 x - 3$ for all values of $x$ .
$g (x) = x^{2} + 4$ for $x ⩾ 0$ .
(a)
Work out the value of $g f (3)$ .
[1 mark]
Answer
(b)
Work out an expression for $g^{- 1} (x)$ .
[2 marks]
Answer
(c)
Hence solve $f g^{- 1} (x) = 7$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
The sketch shows the curve $y = - x^{2} + p x + q$ .
It meets the $x$ -axis at $A (- 1, 0)$ and $B (7, 0)$ . $M$ is the maximum point of the curve.
Not drawn accurately
(a)
Work out the values of $p$ and $q$ .
[2 marks]
Answer $p = \dots\dots q = \dots\dots$
(b)
Hence work out the coordinates of $M$ .
[2 marks]
Answer
(c)
Work out the equation of the line $A M$ . Give your answer in the form $y = m x + c$ .
[2 marks]
Answer
Total for D3: 6 marks
D4
The cost, $£ C$ , of hiring a van for $d$ days is a linear function of $d$ . It costs $£165$ to hire the van for $3$ days and $£325$ to hire it for $7$ days.
(a)
Work out a formula for $C$ in terms of $d$ .
[2 marks]
Answer
(b)
Explain what the gradient of the graph of $C$ against $d$ represents.
[1 mark]
(c)
Ali has $£500$ . Work out the greatest number of days he can hire the van for.
[1 mark]
Answer
Total for D4: 4 marks
D5
The value of a car, $£ V$ , $t$ years after it is bought is modelled by $V = a b^{t}$ , where $a$ and $b$ are constants.
After $1$ year the car is worth $£18 000$ . After $3$ years it is worth $£10 125$ .
(a)
Work out the values of $a$ and $b$ .
[3 marks]
Answer $a = \dots\dots b = \dots\dots$
(b)
After how many complete years is the value of the car first less than $£6000$ ? You must show your working.
[2 marks]
Answer
Total for D5: 5 marks
D6
$f (x) = ⎩ ⎨ ⎧ x + 3, x^{2} + 1, 3 x - 1, - 4 ⩽ x < - 1 - 1 ⩽ x < 1 1 ⩽ x ⩽ 3$
(a)
Draw the graph of $y = f (x)$ on the grid.
[2 marks]
(b)
Work out the value of $f f (- 2)$ .
[1 mark]
Answer
(c)
Solve $f (x) = 0$ . You must show that you have checked each part.
[2 marks]
Answer
(d)
Work out the range of $f$ .
[2 marks]
Answer
Total for D6: 7 marks
D7
The line $L_{1}$ has equation $2 y = 3 x + 8$ . The line $L_{2}$ is parallel to $L_{1}$ and passes through $(4, 1)$ .
(a)
Work out the equation of $L_{2}$ . Give your answer in the form $y = m x + c$ .
[2 marks]
Answer
(b)
$L_{2}$ meets the line $y = x$ at $P$ . Work out the coordinates of $P$ .
[2 marks]
Answer
Total for D7: 4 marks
D8
$f (x) = \frac{12}{x + 2}$ for $0 ⩽ x ⩽ 4$ .
(a)
Work out the range of $f$ .
[2 marks]
Answer
(b)
$g (x) = x^{2} - 4 x$ for all values of $x$ .
Hence work out the range of $g f$ .
[3 marks]
Answer
Total for D8: 5 marks
D9
$f (x) = x^{2} + 10 x + 27$ for $x ⩾ - 5$ .
(a)
Write $f (x)$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer
(b)
Hence write down the range of $f$ .
[1 mark]
Answer
(c)
Work out $f^{- 1} (x)$ and state the domain of $f^{- 1}$ .
[3 marks]
Answer $f^{- 1} (x) = \dots\dots\dots$ domain $\dots\dots\dots$
Total for D9: 6 marks

Extension

Stretch

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
$f (x) = x^{2} - 2 x$ for all values of $x$ , and $g (x) = \frac{1}{x}$ for $x \neq = 0$ .
Work out the range of $f g$ .
[3 marks]
Hint 1 · What to try
Write $f g (x)$ in terms of $x$ , then let $u = \frac{1}{x}$ .
Hint 2 · The first line
$f g (x) = u^{2} - 2 u = (u - 1)^{2} - 1$ . The value of $u$ can be any number except $0$ .
Hint 3 · The full method
$(u - 1)^{2}$ can be any number $⩾ 0$ (it is $0$ when $u = 1$ ). Leaving out $u = 0$ does not lose a value, because $u = 2$ gives the same value as $u = 0$ .
Answer

E2
A straight line with a negative gradient passes through $C (6, 2)$ . It meets the positive $x$ -axis at $A$ and the positive $y$ -axis at $B$ . The area of triangle $O A B$ is $24$ .
Work out the equation of the line. Give your answer in the form $p x + q y = r$ , where $p$ , $q$ and $r$ are integers.
[4 marks]
Not drawn accurately
Hint 1 · What to try
Call the gradient $- m$ , where $m > 0$ . Write the coordinates of $A$ and $B$ in terms of $m$ .
Hint 2 · The first line
$y - 2 = - m (x - 6)$ , so $A = (6 + \frac{2}{m}, 0)$ and $B = (0, 2 + 6 m)$ .
Hint 3 · The full method
$\frac{1}{2} (6 + \frac{2}{m}) (2 + 6 m) = 24$ gives $(6 m + 2)^{2} = 48 m$ , so $(6 m - 2)^{2} = 0$ .
Answer
E3
$f (x) = x^{2} - 8 x + 17$ for $x ⩾ 4$ .
Solve $f (x) = f^{- 1} (x)$ . Give your answer in surd form.
[4 marks]
Hint 1 · What to try
$f$ is increasing for $x ⩾ 4$ , so the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ meet on the line $y = x$ .
Hint 2 · The first line
Solve $x^{2} - 8 x + 17 = x$ , which is $x^{2} - 9 x + 17 = 0$ .
Hint 3 · The full method
$x = \frac{9 \pm 13}{2}$ . Only one of these is in the domain $x ⩾ 4$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use function notation, and find the domain and range of a function.
I can form composite functions and solve equations with them.
I can find inverse functions and state their domains.
I can work with straight-line graphs and find the equation of a line.
I can find the roots, turning point and equation of a quadratic graph.
I can find the two constants of an exponential graph from two points on it, and use exponential models.
I can draw and solve with functions defined in up to three parts.

Answers and mark scheme

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