Functions — Algebra — Grade 9 Mathematics

Understand function notation f(x)
Evaluate functions for given inputs
Find inverse operations with functions
Read function graphs

Input → machine → output

A function is a rule that turns each input into exactly one output. Think of a machine: drop in 3, out comes 7. Drop in 10, out comes 21. The machine here doubles and adds 1.

Definition: function

A function is a rule assigning each input exactly one output. We write f(x) = 2x + 1, read as "f of x equals 2x + 1".

The notation f(x) looks like multiplication but isn't — it's the function's name (f) applied to the input (x).

A function machine: each input has exactly one output.

Evaluating functions

Evaluating f(3) means substituting 3 wherever x appears:

Maths

f(x) = 2x + 1
f(3) = 2(3) + 1 = 7
f(−2) = 2(−2) + 1 = −3
f(0) = 1

Worked example

If g(x) = x² − 4, find g(5) and the values of x for which g(x) = 21.

Maths

g(5) = 25 − 4 = 21
x² − 4 = 21  →  x² = 25  →  x = 5 or x = −5

Going backwards

Functions can be undone by reversing each step. If f(x) = 2x + 1 turns 3 into 7, the reverse turns 7 back into 3: subtract 1, then halve. This reverse function is called the inverse, written f⁻¹(x) = (x − 1)/2.

Not every function has a neat inverse — but every linear function does.

Functions as graphs

Plot input on the x-axis and output on the y-axis, and the function becomes a graph. The graph of f(x) = 2x + 1 is a straight line; reading values off it is the same calculation, done visually. Solving f(x) = 7 means finding where the graph reaches height 7.

One input, one output

The defining rule: each input gives exactly one output. A rule giving two outputs for one input (like "x = ±√y") is not a function. This is the vertical-line test on graphs.

Practice

If f(x) = 3x − 2, find f(4) and f(−1).

Substitute each value for x.

f(4) = 3(4) − 2 = 10. f(−1) = 3(−1) − 2 = −5.

If h(x) = 5 − 2x, find x such that h(x) = 11.

Set up and solve the equation.

5 − 2x = 11 → −2x = 6 → x = −3.

Write the inverse function of f(x) = 3x + 6.

Reverse the steps: what undoes ×3 then +6?

Undo +6 (subtract 6), then undo ×3 (divide by 3): f⁻¹(x) = (x − 6)/3.

Quick check

Functions — quick check

If f(x) = x² + 1, what is f(3)?

f(3) = 3² + 1 = 9 + 1 = 10.

What does f(x) notation mean?

f(x) is read 'f of x' — the output of function f for input x.

If g(x) = 4x, what is the inverse function g⁻¹(x)?

Undo ×4 by ÷4: g⁻¹(x) = x/4.

Key takeaways

A function maps each input to exactly one output; f(x) is "f of x".
Evaluate by substituting the input for x.
Inverse functions reverse the steps: undo last operation first.
Graphs turn functions into pictures — solving becomes reading coordinates.

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