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Piecewise Functions

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen, paper and a calculator handy, to work out your answers.

Question 12 marks

A function f is given by

\begin{aligned}\mathrm{f}(x)&=2x+5 &&x<1\\&=x^2+3 &&1\leqslant x<3\\&=15-2x &&x\geqslant 3\end{aligned}

Work out \mathrm{f}(1)+\mathrm{f}(3)

Hint

At a boundary, look carefully at the inequality signs to decide which piece includes that value of x.

Worked solution
  1. x=1 satisfies 1\leqslant x<3, so use x^2+3: \mathrm{f}(1)=1+3=4
  2. x=3 satisfies x\geqslant3, so use 15-2x: \mathrm{f}(3)=15-6=9
  3. \mathrm{f}(1)+\mathrm{f}(3)=4+9=13

Question 22 marks

A function f is given by

\begin{aligned}\mathrm{f}(x)&=ax^2-3 &&x\leqslant 2\\&=3x+a &&x>2\end{aligned}

a is a constant.

The two parts of the graph of y=\mathrm{f}(x) join where x=2

Work out the value of a.

Hint

The pieces join, so both expressions give the same value when x=2

Worked solution
  1. Substitute x=2 into both pieces.
  2. First piece: 4a-3
  3. Second piece: 6+a
  4. They join, so 4a-3=6+a
  5. 3a=9
  6. a=3

Question 3Challenge6 marks

A function f is given by

\begin{aligned}\mathrm{f}(x)&=4-x &&x<1\\&=x^2+px+q &&1\leqslant x\leqslant 5\\&=2x-15 &&x>5\end{aligned}

p and q are constants.

The three parts of the graph of y=\mathrm{f}(x) join where x=1 and where x=5

A sketch of y=\mathrm{f}(x) is shown.

(a)

Work out the value of p.

2 marks

(b)

Work out the value of q.

1 mark

(c)

Solve \mathrm{f}(x)=-1

Give any non-integer answers to 2 decimal places.

3 marks

Give every value, separated by commas

Hint

Use each linear piece to find the y-coordinate of the joining point, then substitute both joining points into x^2+px+q

Worked solution

Part (a)

  1. At x=1 the first piece gives 4-1=3, so 1+p+q=3
  2. At x=5 the third piece gives 2\times5-15=-5, so 25+5p+q=-5
  3. Subtract the first equation from the second: 24+4p=-8
  4. 4p=-32, so p=-8

Part (b)

  1. Substitute p=-8 into 1+p+q=3
  2. 1-8+q=3
  3. q=10

Part (c)

  1. 4-x=-1 gives x=5, but this piece needs x<1, so reject.
  2. x^2-8x+10=-1 gives x^2-8x+11=0
  3. x=\dfrac{8\pm\sqrt{64-44}}{2}=\dfrac{8\pm\sqrt{20}}{2}=4\pm\sqrt5
  4. 4-\sqrt5=1.763\ldots is in 1\leqslant x\leqslant5 ✓; 4+\sqrt5=6.236\ldots is not, so reject.
  5. 2x-15=-1 gives x=7, which satisfies x>5 ✓
  6. x=1.76 or x=7

More on this topic: Piecewise Functions worksheet with full solutions

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