AQA Level 2 Further Maths · Level 2 Certificate · Chapter 3: Algebra III · Lesson 8

Functions with up to three parts to their domains

About 85 minutesNo calculator59 marks

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Date

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Quick questions to start.

A1
Needed today
Work out the equation of the straight line through $(2, 7)$ and $(6, - 1)$ .
[2 marks]
Answer
A2
Last lesson
The curve $y = a b^{x}$ passes through $(0, 4)$ and $(2, 36)$ , where $a$ and $b$ are positive. Work out $a$ and $b$ .
[2 marks]
Answer
A3
From chapter 3
$f (x) = \frac{x + 4}{5}$ . Work out an expression for $f^{- 1} (x)$ .
[2 marks]
Answer
A4
Needed today
$g (x) = 7 - 2 x$ for $- 1 \leq x \leq 3$ . Work out the range of $g$ .
[2 marks]
Answer

Example, then your turn

Level 2

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

Example 1

Draw the graph of $y = f (x)$ on the grid, where
$f (x) = {2 x + 4, 4, - 2 \leq x < 0 0 \leq x \leq 3$

Hint Work out

f

at each end of each part. The first part is a straight line; the second is a horizontal line.

B1
Draw the graph of $y = f (x)$ on the grid, where
$f (x) = {5, 9 - 2 x, - 1 \leq x < 2 2 \leq x \leq 4$
[3 marks]

Example 2

Draw the graph of $y = f (x)$ on the grid, where
$f (x) = ⎩ ⎨ ⎧ 3, (x - 1)^{2} + 2, 12 - 2 x, - 2 \leq x < 0 0 \leq x < 3 3 \leq x \leq 5$

Hint For the curved part make a table for

x = 0, 1, 2, 3

and join the points with a smooth curve. Check where each part starts and ends.

B2
Draw the graph of $y = f (x)$ on the grid, where
$f (x) = ⎩ ⎨ ⎧ x + 5, 5 - x^{2}, 6 - 2 x, - 3 \leq x < - 1 - 1 \leq x < 1 1 \leq x \leq 3$
[3 marks]

Example 3

The graph of $y = f (x)$ is drawn for $0 \leq x \leq 8$ .

(a) Define $f (x)$ , stating the domain of each part.

(b) State the range of $f$ .

Hint Find the gradient and the

y

-intercept of each straight piece. Each value of

x

belongs to one part only, so use

<

at one end and

\leq

at the other.

B3
The graph of $y = f (x)$ is drawn for $- 2 \leq x \leq 6$ .
(a)
Define $f (x)$ , stating the domain of each part.
[3 marks]
(b)
State the range of $f$ .
[1 mark]
Answer
Total for B3: 4 marks

Example 4

$f (x) = ⎩ ⎨ ⎧ 3 x + 2, x^{2} - 6 x + 2, 3 x - 16, x < 0 0 \leq x \leq 6 x > 6$

Solve $f (x) = - 3$ .

Hint Solve

f (x) = - 3

for each part on its own. Then keep only the answers that lie in that part's domain.

B4
$f (x) = ⎩ ⎨ ⎧ 2 x + 1, x^{2} - 4 x + 1, 21 - 3 x, x < 0 0 \leq x \leq 5 x > 5$
Solve $f (x) = - 2$ .
[4 marks]
Answer $x = \dots\dots\dots$

Practice

Level 2

C1
$f (x) = ⎩ ⎨ ⎧ 3 x, 5 - 2 x, x - 7, 0 \leq x < 1 1 \leq x < 4 4 \leq x \leq 7$
The sketch shows $y = f (x)$ . The graph meets the $x$ -axis at $x = 0$ , $x = a$ and $x = 7$ .
Not drawn accurately
(a)
Work out the value of $a$ .
[1 mark]
Answer $a = \dots\dots$
(b)
Work out the ratio of the area above the $x$ -axis to the area below it, between the graph and the $x$ -axis. Give your answer in its simplest form.
[3 marks]
Answer $\dots\dots : \dots\dots$
Total for C1: 4 marks
C2
A walker sets off from home. Her distance from home, $d$ km, after $t$ hours is
$d = ⎩ ⎨ ⎧ 4 t, 6, 12 - 3 t, 0 \leq t < 1.5 1.5 \leq t < 2 2 \leq t \leq 4$
(a)
Describe what the walker does between $t = 1.5$ and $t = 2$ .
[1 mark]
(b)
Work out her speed on the way home.
[1 mark]
Answer $\dots\dots$ km/h
(c)
Work out the two times when she is $3$ km from home.
[2 marks]
Answer $t = \dots\dots$ and $t = \dots\dots$
Total for C2: 4 marks

C3
$f (x) = {x^{2} + k, 2 x + 5, 0 \leq x < 3 3 \leq x \leq 6$
The two parts of the graph of $y = f (x)$ join at $x = 3$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
C4
$f (x) = ⎩ ⎨ ⎧ 2 - x, x^{2} + 2, 10 - 2 x, - 3 \leq x < 0 0 \leq x < 2 2 \leq x \leq 4$
Work out the range of $f$ .
[2 marks]
Answer

Exam-style questions

AQA exam style

D1
$f (x) = {x^{2}, 15 - x, x < 4 x \geq 4$
Circle the value of $f (4)$ .
$7111216$
[1 mark]
Answer

D2
A function $f$ is given by
$f (x) = ⎩ ⎨ ⎧ x + 6, 8 - x^{2}, 3 x - 10, x < - 2 - 2 \leq x \leq 3 x > 3$
A sketch of $y = f (x)$ is shown. Work out all the values of $x$ for which $f (x) = 0$ . Give your answers in exact form. You must show your working.
[5 marks]
Not drawn accurately
Answer $x = \dots\dots\dots$
D3
$f (x) = {(x - a)^{2} + b, c x + d, 0 \leq x < 3 3 \leq x \leq 6$
$a$ , $b$ , $c$ and $d$ are constants. The sketch shows $y = f (x)$ . The curve has its minimum point at $(2, - 1)$ , the graph ends at $(6, 9)$ , and the two parts join at $x = 3$ . Work out the values of $a$ , $b$ , $c$ and $d$ .
[4 marks]
Not drawn accurately
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$ $d = \dots\dots$
D4
$f (x) = ⎩ ⎨ ⎧ 2 x + 8, x^{2} + 5, 13 - 2 x, - 4 \leq x < - 1 - 1 \leq x < 2 2 \leq x \leq 5$
Priya says, "The equation $f (x) = 4$ has three solutions, one from each part."
Priya is wrong.
(a)
Show that $f (x) = 4$ has exactly two solutions, and work them out.
[3 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$
(b)
Work out the range of $f$ .
[2 marks]
Answer
Total for D4: 5 marks
D5
A train starts from rest and speeds up at a steady rate for $40$ seconds until its speed is $20$ m/s. It then travels at $20$ m/s for $T$ seconds. It then slows down at a steady rate for $50$ seconds until it stops. The train travels $4500$ m in total.
(a)
Work out the value of $T$ .
[3 marks]
Answer $T = \dots\dots$
(b)
The speed of the train is $v$ m/s after $t$ seconds. Write $v$ as a function of $t$ , stating the domain of each part.
[3 marks]
Total for D5: 6 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$f (x) = ⎩ ⎨ ⎧ x + 3, x^{2} - 4 x + 3, 11 - 2 x, - 3 \leq x < 0 0 \leq x < 4 4 \leq x \leq 6$
Work out the values of $k$ for which the equation $f (x) = k$ has exactly three solutions.
[4 marks]
Hint 1 · What to try
Sketch $y = f (x)$ , then imagine the horizontal line $y = k$ moving up and down.
Hint 2 · The first line
The middle part is $(x - 2)^{2} - 1$ : its lowest point is $(2, - 1)$ . Each part runs from $y = 3$ (at one end) down to its lowest value.
Hint 3 · The full method
For $k < - 1$ there are no solutions; $k = - 1$ gives two; for $- 1 < k < 0$ the curve gives two and the last part one; for $0 \leq k < 3$ the first part adds one more, making four. So $- 1 < k < 0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw the graph of a function defined in two or three parts.
I can define a function from its graph, giving the domain of each part, and state its range.
I can solve $f (x) = k$ , keeping only the solutions that lie in their own part.
I can use speed-time and distance-time functions in context.

Answers and mark scheme

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