Functions — Mathslive.ie

Section 3 of 6

Solving — finding x

Working backwards

Given the output, find the input: set the rule equal to the given value and solve like an equation.

Worked example — a clean integer answer

Given $f(x) = 2x - 1$, find $f(3)$ and find $x$ when $f(x) = 19$.

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

$2x - 1 = 19$

$2x = 20$

$x = 10$

$f(3) = 5,\ x = 10$

You try

Given $g(x) = 3x - 5$, find $g(7)$ and find $x$ when $g(x) = 19$.

Sub in for $g(7)$. Then set $3x - 5 = 19$ and solve.

$g(7) = 3(7) - 5 = 16$

$3x - 5 = 19$

$3x = 24$

$x = 8$

$g(7) = 16,\ x = 8$

Worked example — when the answer is a fraction

Given $f(x) = 7x - 3$, find $f(9)$ and find $x$ when $f(x) = 43$.

$f(9) = 7(9) - 3 = 60$

$7x - 3 = 43$

$7x = 46$

$x = \tfrac{46}{7}$

$f(9) = 60,\ x = \tfrac{46}{7}$

Leave it exact

Don’t force a whole number. Leave $\tfrac{46}{7}$ as a fraction — that’s the exact answer. Decimals lose marks for rounding when the question hasn’t asked.

When $f(x) = g(\text{a number})$

Worked example — compute g first, then solve

Given $f(x) = 2x + 3$ and $g(x) = 5x - 1$, find $x$ when $f(x) = g(3)$.

$g(3) = 5(3) - 1 = 14$

$2x + 3 = 14$

$2x = 11$

$x = \tfrac{11}{2}$

You try

Given $f(x) = 5x - 2$ and $g(x) = 3x + 1$, find $x$ when $f(x) = g(9)$.

Work out $g(9)$ first — it’s just a number. Then solve $5x - 2 = $ that.

$g(9) = 3(9) + 1 = 28$

$5x - 2 = 28$

$5x = 30$

$x = 6$

When $f(x) = g(x)$

Worked example — two linear rules

Given $f(x) = 5x - 3$ and $g(x) = 3x + 9$, find $x$ when $f(x) = g(x)$.

$5x - 3 = 3x + 9$

$5x - 3x = 9 + 3$

$2x = 12$

$x = 6$

You try

Given $f(x) = 6x - 7$ and $g(x) = 3x + 8$, find $x$ when $f(x) = g(x)$.

Set the two rules equal, gather the $x$’s on one side.

$6x - 7 = 3x + 8$

$6x - 3x = 8 + 7$

$3x = 15$

$x = 5$

Find the missing letter

Worked example — find k

Given $g(x) = 3x + k$ and $g(5) = 20$, find $k$.

$3(5) + k = 20$

$15 + k = 20$

$k = 5$

You try

Given $f(x) = ax + 9$ and $f(3) = 36$, find $a$.

Sub the pair in: $a(3) + 9 = 36$, then solve for $a$.

$3a + 9 = 36$

$3a = 27$

$a = 9$

Part 3 done.

Next up: Composite Functions. Head back to the hub for the next part.