Functions — Mathslive.ie

Section 1 of 6

The Basics — input, rule, output

A function takes an input, runs it through a rule, and gives you an output. Same input, same rule — same output, every single time.

The three parts

The input is $x$. The rule is $f$ (or $g$, $h$). The set of inputs is the domain; the set of outputs is the range.

$f(x)$ is shorthand for ‘put $x$ into the rule $f$’. To find $f(3)$, put $3$ wherever you see $x$ — always in brackets.

Worked example — three quick evaluations

Given $f(x) = 3x + 1$, find $f(1)$, $f(2)$ and $f(3)$.

$f(1) = 3(1) + 1 = 4$

$f(2) = 3(2) + 1 = 7$

$f(3) = 3(3) + 1 = 10$

$4,\ 7,\ 10$

Worked example — a quadratic rule

Given $f(x) = x^2 + 3x$, evaluate at $x = -1, 0, 1, 2, 3$.

$f(-1) = (-1)^2 + 3(-1) = 1 - 3 = -2$

$f(0) = 0$

$f(1) = 4$

$f(2) = 10$

$f(3) = 18$

$-2,\ 0,\ 4,\ 10,\ 18$

Watch the negatives

$(-1)^2$ is $+1$, not $-1$. Squaring kills the minus sign — the single most-failed step on this topic.

You try

Given $g(x) = 2x - 1$, find $g(2)$, $g(5)$ and $g(-7)$.

Put each number in, in brackets. Mind the sign on $-7$.

$g(2) = 2(2) - 1 = 3$

$g(5) = 2(5) - 1 = 9$

$g(-7) = 2(-7) - 1 = -15$

$3,\ 9,\ -15$

A couple is the pair $(x, f(x))$ — the point you’d plot on a graph.

Worked example — forming couples

Given $f(x) = x^2 + 3x$, form the couples for the domain $\{-1,0,1,2,3\}$.

$f(-1) = -2 \Rightarrow (-1, -2)$

$f(0) = 0 \Rightarrow (0, 0)$

$f(1) = 4 \Rightarrow (1, 4)$

$f(2) = 10 \Rightarrow (2, 10)$

$f(3) = 18 \Rightarrow (3, 18)$

$(-1,-2),(0,0),(1,4),(2,10),(3,18)$

You try

For $g(x) = 2x - 1$, write the couples for inputs $2, 5, -7$.

You already found the outputs: $3, 9, -15$. Pair each input with its output.

$(2, 3),\ (5, 9),\ (-7, -15)$

$(2,3),(5,9),(-7,-15)$

Worked example — two functions

Let $f(x) = 5x - 3$ and $g(x) = 3x + 9$. Find $f(1)$ and $g(5)$.

$f(1) = 5(1) - 3 = 2$

$g(5) = 3(5) + 9 = 24$

$f(1) = 2,\ g(5) = 24$

Part 1 done.

Next up: Domain, Range & Arrow Diagrams. Head back to the hub for the next part.