Functions

A function machine takes an input, does operations to it in order, and gives an output.

• To go forwards, do the operations in order.
• To go backwards, do the inverse operations in reverse order.

Example: a machine does input → × 3 → − 4 → output. Find the output for an input of 5, and the input for an output of 20.

1. Forwards: 5 × 3 = 15, then 15 − 4 = 11
2. Backwards from 20: undo − 4 by adding 4: 24
3. Undo × 3 by dividing by 3: 8

With an input of x, the output is 3x − 4. A function machine can turn words into an expression.

f(x) is a function's output for an input of x. If f(x) = 3x − 4, then f(5) means "put 5 in": f(5) = 3 × 5 − 4 = 11.

1. To evaluate f(a), substitute a for x.
2. To solve f(x) = k, form the equation and solve for x.

Example: f(x) = x² + 1. Find f(−3), and solve f(x) = 26.

1. f(−3) = (−3)² + 1 = 9 + 1 = 10
2. x² + 1 = 26
3. x² = 25
4. x = 5 or x = −5

A composite function applies one function after another. fg(x) means "do g first, then f": f(g(x)).

1. Work out the inner function, g, first.
2. Put its answer (or expression) into f.

Example: f(x) = 3x − 4 and g(x) = x². Find fg(2), gf(2) and fg(x).

1. fg(2): g(2) = 4, then f(4) = 3 × 4 − 4 = 8
2. gf(2): f(2) = 2, then g(2) = 4
3. fg(x) = f(x²) = 3x² − 4

The inverse function f⁻¹(x) undoes f: if f(5) = 11, then f⁻¹(11) = 5.

1. Write y = f(x).
2. Rearrange to make x the subject.
3. Swap y for x: that gives f⁻¹(x).

Example: f(x) = 3x − 4. Find f⁻¹(x).

1. y = 3x − 4
2. y + 4 = 3x, so x = (y + 4)/3
3. f⁻¹(x) = (x + 4)/3

Check: f⁻¹(11) = 15/3 = 5, and f(5) = 11.

Harder example: f(x) = (2x + 1)/5. Find f⁻¹(x).

1. y = (2x + 1)/5, so 5y = 2x + 1
2. 5y − 1 = 2x, so x = (5y − 1)/2
3. f⁻¹(x) = (5x − 1)/2