Functions

**Temperature converter.** A function converts temperature from Celsius to Fahrenheit: $F(c) = \frac{9}{5}c + 32$. (a) Find $F(0)$ and $F(100)$. (b) Find $F^{-1}(F)$, the inverse function (Fahrenheit to Celsius). (c) Find $F^{-1}(212)$. Interpret. (d) Find the temperature where Celsius and Fahrenheit are equal: $F(c) = c$.

**Function machines.** Given $f(x) = 3x - 1$ and $g(x) = x^2 + 2$: (a) Find $f(2)$, $g(2)$, $f(g(2))$, and $g(f(2))$. (b) Find a formula for $(f \circ g)(x)$ and $(g \circ f)(x)$. (c) Solve $(f \circ g)(x) = 14$.

**Inverse practice.** Find the inverse of each function. State any domain restrictions needed. (a) $f(x) = 5x - 7$ (b) $g(x) = \dfrac{2x + 3}{4}$ (c) $h(x) = (x - 2)^2$ (for $x \geq 2$)

**Modelling a fence.** A farmer has 60 m of fencing to enclose a rectangular field, one side of which uses an existing wall (so no fencing on that side). If the side perpendicular to the wall has length $x$ metres: (a) Express the side along the wall in terms of $x$. (b) Express the area $A(x)$ as a function of $x$. (c) State the domain of $A$. (d) Find the value of $x$ that maximises the area.

**Composition puzzle.** $f$ and $g$ are linear functions with $f(x) = 2x + 1$. If $(f \circ g)(x) = 4x + 7$, find $g(x)$.

**Identifying functions.** For each relation, decide whether it represents a function. Justify. (a) Each Year 10 student maps to their unique school ID. (b) Each town maps to all citizens who live there. (c) Each $x \in \mathbb{R}$ maps to its square $x^2$. (d) Each positive number $y$ maps to all $x$ with $x^2 = y$.

**Domain and range.** A function is defined by $f(x) = \dfrac{1}{x - 3}$. (a) State the largest possible domain. (b) Find the range. (c) Find $f^{-1}(x)$ and its domain.

**Sketch and interpret.** A function has graph passing through $(0, 4)$, $(2, 0)$, $(3, -1)$, $(5, 1)$, $(6, 4)$. (a) Estimate the range from the data. (b) Is the function one-to-one over $[0, 6]$? Justify. (c) Why does this matter for finding an inverse?

**Currency converter.** £1 = €1.18 (May 2026 rate). (a) Write a function $E(p)$ converting £$p$ to euros. (b) Find $E^{-1}$ and interpret. (c) Tomás is travelling from Geneva to London with €500. What does he have in pounds (2 d.p.)?

**Restricted-domain inverses.** Consider $f(x) = x^2$ on the full domain $\mathbb{R}$. (a) Why does $f$ not have an inverse on this domain? (b) Restrict to $x \geq 0$. Now what is $f^{-1}$? (c) Restrict to $x \leq 0$. Now what is $f^{-1}$? (d) Verify (b): compute $(f \circ f^{-1})(9)$ and $(f^{-1} \circ f)(3)$.

**Self-inverse functions.** A function $f$ is *self-inverse* if $f^{-1}(x) = f(x)$ for all $x$. (a) Show that $f(x) = -x$ is self-inverse. (b) Show that $f(x) = \dfrac{1}{x}$ (for $x \neq 0$) is self-inverse. (c) Find all linear functions of the form $f(x) = ax + b$ that are self-inverse.

**A composition mystery.** Given $f(x) = 2x + 5$ and $h(x) = 4x + 13$, find a function $g$ such that $(f \circ g)(x) = h(x)$. Also state whether $g$ is unique. Justify.