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Functions

Functions

Function notation: evaluating and solving

Type 1 Extended 5 min read

On this page
  1. In this note
  2. Formula bank
  3. A function is a machine
  4. Evaluating
  5. Solving with functions
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 2.12 (4024) / 2.13 (0580): function notation Extended
  • 2.12Use f(x) notation: a function gives exactly one output for each input.
  • 2.12Evaluate \(f(5)\), \(f(-2)\) and expressions such as \(f(x-2)\), \(g(2x)\).
  • 2.12Solve \(f(x)=25\), \(f(x)=g(x)\) and inequalities like \(f(x)<13\).

Before you start: Substitution and expanding. Next: Inverse functions.

Formula bank

EvaluateLearn it\(f(a)\): replace every \(x\) by \((a)\)a = a number or an expression; keep the brackets.Use it when 'find f(3)', 'find f(x − 2)'.
SolveLearn it\(f(x)=k\): solve 'the rule = k'the OUTPUT is given; find the input.Use it when 'solve f(x) = …'.

A function is a machine

\(f(x)=4x-3\) means: the function \(f\) multiplies its input by 4 and subtracts 3. \(x\) just stands for 'whatever goes in'. So \(f(5)=4(5)-3=17\): put 5 in, 17 comes out.

Function machine for f(x) = 4x − 35× 4− 317
f(5) = 17: put 5 in, 17 comes out.
\(f(5)\) is the OUTPUT for input 5. It is not \(f\times5\).

Evaluating

General methodHow to evaluate a function

  1. Write the rule again with every \(x\) replaced by the input IN BRACKETS.
  2. A number: work it out (BIDMAS; square negatives in brackets).
  3. An expression (like x − 2 or 2x): expand and simplify.

Example \(f(x)=4x-3\). Find \(f(x-2)\) in its simplest form.

Replace \(x\) by \((x-2)\), then expand.

  1. \(4(x-2)-3\)
  2. \(=4x-8-3\)

Answer \(4x-11\)

Try it yourself

1
\(f(x)=4x-3\). Find \(f(-2)\).
Show answer

Answer −11

2
\(g(x)=x^{2}+1\). Find \(g(-3)\).
Show answer

Answer 10

3
\(g(x)=x^{2}+1\). Find \(g(2x)\).
Hint
\((2x)^{2}=4x^{2}\).
Show answer

Answer \(4x^{2}+1\)

Solving with functions

General methodHow to solve an equation with functions

  1. Write the rule(s) out: \(f(x)=k\) becomes 'rule = k'; \(f(x)=g(x)\) becomes 'rule = rule'.
  2. Solve as a normal equation (a squared x gives two answers).
  3. For an inequality, solve it the same way and keep the sign.

Example \(f(x)=4x-3\). Solve \(f(x)=2x+9\).

Set the rule equal to 2x + 9.

  1. \(4x-3=2x+9\)
  2. \(2x=12\)

Answer \(x=6\)

Try it yourself

1
\(f(x)=4x-3\). Solve \(f(x)=25\).
Show answer

Answer 7

2
\(g(x)=x^{2}+1\). Solve \(g(x)=50\).
Show answer

Answer 7, −7

Ways they test it

Type 1 Find f(number)

"Find \(f(-8)\)." (1 mark)

Spot it a number in the brackets.

  1. Substitute in brackets.
  2. BIDMAS.

Type 2 Find f(expression)

"Find \(f(x-2)\) in its simplest form." (2 marks)

Spot it an expression in the brackets.

  1. Replace x by the whole bracket.
  2. Expand and simplify.

For example \(f(x)=4x-3\)

  1. \(4(x-2)-3\)

Answer \(4x-11\)

Type 3 Solve f(x) = k

"Solve \(k(x)=70\)" with \(k(x)=3x^{2}-5\). (2 marks)

Spot it 'solve' with an output given.

  1. Rule = k.
  2. Solve (± for a square).

For example \(3x^{2}-5=70\)

  1. \(x^{2}=25\)

Answer \(x=\pm5\)

Type 4 Fraction rule

"\(h(x)=\dfrac{2x+1}3\). Solve \(h(x)=x-4\)." (2 marks)

Spot it a rule with a denominator.

  1. Multiply both sides by the denominator.
  2. Solve.

For example \(2x+1=3(x-4)\)

  1. \(2x+1=3x-12\)

Answer \(x=13\)

Mistakes to avoid

  • ✗ \(g(2x)=2x^{2}+1\)✓ \(4x^{2}+1\)Square the whole input.
  • ✗ \(f(x-2)=4x-5\)✓ \(4x-11\)The input changes BEFORE the rule acts.
  • ✗ \(g(x)=50\Rightarrow x=7\)✓ \(x=\pm7\)Two square roots.
  • ✗ \(k(-1)=3(-1)^{2}-5=-8\)✓ −2\((-1)^{2}=+1\).

Summary

  • f(a) = the output when a goes in: replace x by (a).
  • f(x) = k: solve rule = k.
  • f(expression): substitute, then expand and simplify.
  • One input → exactly one output.

If you see …

  • "find f(…)"substitute.
  • "solve f(x) = …"make an equation and solve it.

Practice

Practice: easy, medium, hard

Easy

1
\(f(x)=4x-3\). Find \(f(0)\).
Show answer

Answer −3

2
\(m(x)=7-2x\). Find \(m(-4)\).
Show answer

Answer 15

3
\(f(x)=4x-3\). Solve \(f(x)=9\).
Show answer

Answer 3

Medium

1
\(g(x)=x^{2}+1\). Find \(g(x-1)\) in its simplest form.
Guide me

What is it asking? an expression

What do we know? replace x by \((x-1)\)

Plan expand the square

Show working
  1. \((x-1)^{2}+1\)
  2. \(x^{2}-2x+1+1\)

Answer \(x^{2}-2x+2\)

2
\(h(x)=\dfrac{2x+1}3\). Solve \(h(x)=x-4\).
Guide me

What is it asking? x

What do we know? × 3

Plan \(2x+1=3x-12\)

Show working
  1. \(2x+1=3x-12\)
  2. \(x=13\)

Answer 13

3
\(k(x)=3x^{2}-5\). Solve \(k(x)=70\).
Guide me

What is it asking? x

What do we know? \(3x^{2}-5=70\)

Plan \(x^{2}=25\)

Show working
  1. \(3x^{2}=75\)
  2. \(x=\pm5\)

Answer 5, −5

Hard

1
\(f(x)=4x-3\). Show that \(f(x+1)-f(x-1)\) is a constant and find it.
Guide me

What is it asking? a number

What do we know? \(f(x+1)=4x+1\)

Plan \(f(x-1)=4x-7\)

Show working
  1. \((4x+1)-(4x-7)=8\)

Answer 8

PartAnswer / workingMarksGuidance
both expressionsM1
8A1
Total: 2 marks
2
\(g(x)=x^{2}+1\). Simplify \(g(x+3)-g(x)\).
Guide me

What is it asking? an expression

What do we know? \(g(x+3)=x^{2}+6x+10\)

Plan subtract \(x^{2}+1\)

Show working
  1. \(x^{2}+6x+10-x^{2}-1\)
  2. \(6x+9\)

Answer \(6x+9\)

PartAnswer / workingMarksGuidance
\(x^{2}+6x+10\)M1
\(6x+9\)A1
Total: 2 marks
3
\(f(x)=4x-3\). Find the largest integer \(x\) with \(f(x)<13\).
Guide me

What is it asking? an integer

What do we know? \(4x-3<13\)

Plan \(x<4\)

Show working
  1. \(4x<16\)
  2. \(x<4\)
  3. largest integer 3

Answer 3

PartAnswer / workingMarksGuidance
\(x<4\)M1
3A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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