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Learn · Mathematics · Grade 10 · Functions

Functions: notation, graphs and linear functions

What makes a relation a function, f(x) notation, domain and range, the vertical line test, and the graph of f(x) = mx + n read from slope and intercept.

High school Lesson 16 of 31 Interactive figure Practice

First, the rules

One input, one output

★ What a function is
Example 1
A rule that sends every element of the domain A to exactly one element of the codomain B: f : A → B. The set of outputs actually hit is the range (image).
Notation
Example 1
f(x) = 2x + 1 ⇒ f(3) = 7, f(a + 1) = 2a + 3
Replace every x by the input, brackets and all.
Vertical line test
A graph is a function if no vertical line crosses it twice.
Even and odd
f(−x) = f(x) even (mirror in the y-axis) · f(−x) = −f(x) odd (symmetric through the origin)
★ Composition
Example 2
(f ∘ g)(x) = f(g(x))
Inside first. In general f ∘ g ≠ g ∘ f.
Inverse
Example 2
Swap x and y and solve for y. Only one-to-one functions have inverses; the graph of f⁻¹ is the mirror of f in the line y = x.
★ Linear function
Example 3
f(x) = mx + n
m = slope = rise / run; n = y-intercept. Parallel lines share m. Through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁).
Domain of a formula
No division by zero, no square root of a negative: for 1 / (x − 2) exclude x = 2; for √(x − 3) need x ≥ 3.

See it

Slope and intercept

Move m and n. The small step shows "one right, m up". Negative m tilts the line down; n slides it up and down without turning it.

Step by step

Worked examples

1

Evaluating and solving

notation

f(x) = x² − 3x + 1. Find f(−2), f(a + 1), and the values of x with f(x) = 1.

Ask yourself: Every x gets replaced by the same thing — in brackets.

Show the solution
  1. f(−2) = (−2)² − 3(−2) + 1 = 4 + 6 + 1 = 11.
  2. f(a + 1) = (a + 1)² − 3(a + 1) + 1 = a² + 2a + 1 − 3a − 3 + 1 = a² − a − 1.
  3. x² − 3x + 1 = 1 ⇒ x(x − 3) = 0 ⇒ x = 0 or x = 3.

Answer 11; a² − a − 1; x ∈ {0, 3}

2

Composition and inverse

f ∘ g, f⁻¹

f(x) = 2x − 5 and g(x) = x² + 1. Find (f ∘ g)(2), (g ∘ f)(2) and f⁻¹(x).

Ask yourself: Inside first: g(2) before f. For the inverse, swap the letters.

Show the solution
  1. g(2) = 5, f(5) = 5, so (f ∘ g)(2) = 5.
  2. f(2) = −1, g(−1) = 2, so (g ∘ f)(2) = 2 — different order, different answer.
  3. Inverse: y = 2x − 5 → swap → x = 2y − 5 → y = x + 52. Check: f⁻¹(f(4)) = f⁻¹(3) = 4 ✓.

Answer 5; 2; f⁻¹(x) = x + 52

3

The line through two points

linear function

A linear function has f(1) = 4 and f(3) = 10. Find f(x) and the point where its graph crosses the x-axis.

Ask yourself: Two points give the slope; one point then gives n.

Show the solution
  1. Slope: m = 10 − 43 − 1 = 3.
  2. Through (1, 4): 4 = 3 · 1 + n ⇒ n = 1, so f(x) = 3x + 1.
  3. x-intercept where f(x) = 0: 3x + 1 = 0 ⇒ x = −13.

Answer f(x) = 3x + 1; crosses at (−13, 0)

Practice: functions

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Evaluate, compose, invert, find a slope or a domain — numbers typed, forms chosen. Your best score is kept in this browser only.

About this lesson

Functions for high school: the definition (each input, one output), f(x) notation and evaluating f(2) or f(a + 1), domain and range, the vertical line test, even and odd functions, composition f(g(x)) and inverse functions, and linear functions f(x) = mx + n — slope as rise over run, the y-intercept, parallel lines, and the line through two points. Move the slope and intercept yourself, then practise.

Part of Learn mathematics · Grade 10 · Functions. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.