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Combining Functions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Sum, difference, product, quotient: $(f\pm g)(x)=f(x)\pm g(x)$ and $(fg)(x)=f(x)\,g(x)$ . The domain is where both $f$ and $g$ are defined; for $\dfrac{f}{g}$ , also remove every $x$ with $g(x)=0$ .
Composition: $(f\circ g)(x)=f(g(x))$ : apply $g$ first, then $f$ . In general $f(g(x))\ne g(f(x))$ , and $f\circ g$ is not the product $fg$ .
Domain of $f\circ g$ : $x$ must be in the domain of $g$ and $g(x)$ must be in the domain of $f$ . Find it before you simplify.
Inequalities: move everything to one side, factor, and test every interval. A squared factor does not change sign, and never divide by an expression whose sign you don't know.
Squaring both sides can create extraneous roots: check every solution in the original equation.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Sums & Differences · Products & Quotients · Composition · Solving

K1 Sums & Differences

a)
For $f(x)=3x^2-2x+5$ and $g(x)=x^2-4x+7$ , simplify $(f-g)(x)$ .
b)
For $p(x)=\sqrt{x+2}$ and $q(x)=\sqrt{4-x}$ , state the domain of $(p+q)(x)$ in interval notation.
c)
The lines $y=f(x)$ and $y=g(x)$ are graphed below. Each grid square is 1 unit. Evaluate $(f+g)(4)$ , then find the value of $x$ where $(f+g)(x)=0$ .

Stuck? Peek at a hint

Put brackets around $g(x)$ before subtracting, so the minus sign reaches every term. For the graph, read $f(4)$ and $g(4)$ separately, then add.

K2 Products & Quotients

a)
For $f(x)=x+4$ and $g(x)=2x-3$ , simplify $(fg)(x)$ .
b)
For $f(x)=x^2-9$ and $g(x)=x^2+x-6$ , simplify $\left(\dfrac{f}{g}\right)(x)$ and state its domain.
c)
For $u(x)=\sqrt{x-1}$ and $v(x)=x-4$ , state the domains of $\left(\dfrac{u}{v}\right)(x)$ and $\left(\dfrac{v}{u}\right)(x)$ in interval notation.

Stuck? Peek at a hint

Find the domain from the original $f$ and $g$ before you cancel anything. Then remove every $x$ that makes the denominator function zero.

K3 Composition from a Table

Use the table of values for $f$ and $g$ .

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$3$	$0$	$-1$	$0$	$3$	$8$
$g(x)$	$2$	$3$	$1$	$-2$	$0$	$-1$

a)
Evaluate $f(g(0))$ and $g(f(0))$ .
b)
Evaluate $(g\circ g)(1)$ and $(f\circ g)(2)$ .
c)
Find every $x$ in the table for which $f(g(x))=0$ .

Stuck? Peek at a hint

Work from the inside out: find the inner value $g(a)$ in the table first, then look up that number in the $x$ row again.

K4 Composition Rules

a)
For $f(x)=x^2+x$ and $g(x)=x-2$ , simplify $f(g(x))$ .
b)
For the same $f$ and $g$ , simplify $g(f(x))$ .
c)
For $p(x)=x^2-5$ and $q(x)=\sqrt{x+3}$ , simplify $p(q(x))$ and state its domain in interval notation.

Stuck? Peek at a hint

Replace every $x$ in the outer function by the whole inner expression, in brackets.

K5 Solving with Combined Functions

a)
Find the points of intersection of $f(x)=x^2-2x$ and $g(x)=x+4$ .
b)
For $p(x)=\sqrt{x+6}$ and $q(x)=x$ , solve $p(x)=q(x)$ .
c)
For $f(x)=(x+1)^2$ and $g(x)=x-3$ , solve $(fg)(x)>0$ . Give your answer in interval notation.

Stuck? Peek at a hint

Set the two expressions equal (or move everything to one side) and solve. Then substitute each answer, or a test point from each interval, into the original.

Real World (A)

Profit Models · Damped Motion · Composite Models

A1 Festival Food Truck

A food truck at a summer festival in Ottawa sells $x$ meals a day when it charges $p(x)=15-0.025x$ dollars per meal. Its daily cost is $C(x)=4x+400$ dollars.

a)
Write the revenue $R(x)=x\cdot p(x)$ and the profit $P(x)=R(x)-C(x)$ in simplified form.
b)
How many meals must the truck sell to break even? For which numbers of meals does it make a profit?
c)
The truck's fridge holds ingredients for only 150 meals a day. How many meals should it sell to make the most profit? State the price per meal and the daily profit.

Stuck? Peek at a hint

Multiply out $x(15-0.025x)$ , then subtract all of $C(x)$ in brackets.

A2 Diving Board

After a diver leaves a diving board, the vertical position of the board's tip, in centimetres above its rest position, is modelled by $d(t)=12(0.5)^t\cos(2\pi t)$ , where $t$ is the time in seconds.

a)
Evaluate $d(0.5)$ and $d(1)$ , to one decimal place. Where is the tip at each time?
b)
The factor $12(0.5)^t$ sets the size of each swing. After how many seconds are the swings smaller than 0.5 cm? Round to one decimal place.
c)
Evaluate the swing size $12(0.5)^t$ at $t=10$ . According to the model, does the board ever stop moving? Is the model still useful at $t=10$ ?

Stuck? Peek at a hint

For (b), solve $12(0.5)^t=0.5$ using logarithms.

A3 Weather Balloon

A weather balloon rises so that its altitude, in kilometres, $t$ minutes after launch is $h(t)=0.3t$ . Up to an altitude of 11 km, the air temperature in °C at altitude $h$ km is $T(h)=15-6.5h$ .

a)
Find $T(h(t))$ in simplified form. What do its input and its output represent?
b)
When does the balloon reach air at $-24$ °C, and at what altitude?
c)
Evaluate $T(h(50))$ . Should you believe the result? State the domain of $T\circ h$ in this context.

Stuck? Peek at a hint

The balloon's altitude depends on time, and the temperature depends on altitude: $h$ is the inner function.

Think It Through (T)

Error Analysis · Building Functions

T1 Spot the Flaw

A student solves $f(x)\ge g(x)$ for $f(x)=x^3$ and $g(x)=4x$ :
" $x^3\ge4x$ . Divide both sides by $x$ : $x^2\ge4$ . Therefore $x\le-2$ or $x\ge2$ ."

a)
Test $x=-1$ and $x=-3$ in the original inequality. What does each test show about the student's answer?
b)
Identify the flawed step and explain what went wrong.
c)
Solve $x^3\ge4x$ correctly. Give your answer in interval notation.

Stuck? Peek at a hint

For (c), write $x^3-4x\ge0$ and factor completely.

T2 Build It Backwards

Many answers are possible. Neither $f$ nor $g$ may be $y=x$ .

a)
Find $f$ and $g$ such that $f(g(x))=(3x-1)^5+2$ .
b)
Write $k(x)=\sqrt{x^2+1}$ as $f(g(x))$ in two different ways.
c)
Find two functions $f$ and $g$ with $f(g(x))=g(f(x))$ for every $x$ . Show that they work.

Stuck? Peek at a hint

Ask what is done to $x$ first: that step is the inner function $g$ .

Boss Level

Optional challenge

Going in Circles

Let $f(x)=\dfrac{1}{1-x}$ . Define $f_1(x)=f(x)$ and $f_{n+1}(x)=f(f_n(x))$ , so $f_n$ applies $f$ a total of $n$ times.

a)
Simplify $f_2(x)=f(f(x))$ .
b)
Simplify $f_3(x)$ .
c)
Find $f_{2026}(2)$ .

Stuck? Peek at a hint

For (a), write $1-\dfrac{1}{1-x}$ as a single fraction first. For (c), apply $f$ to $2$ a few times and watch for a repeat.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $(f-g)(x)=2x^2+2x-2$
b) $[-2,4]$
c) $(f+g)(4)=3$ ; $x=-2$

a) $(fg)(x)=2x^2+5x-12$
b) $\left(\dfrac{f}{g}\right)(x)=\dfrac{x-3}{x-2}$ ; $D=\{x\in\mathbb{R}\mid x\ne-3,\ x\ne2\}$
c) $\dfrac{u}{v}$ : $[1,4)\cup(4,\infty)$ ; $\dfrac{v}{u}$ : $(1,\infty)$

a) $f(g(0))=0$ ; $g(f(0))=3$
b) $(g\circ g)(1)=2$ ; $(f\circ g)(2)=-1$
c) $x=0$ and $x=3$

a) $f(g(x))=x^2-3x+2$
b) $g(f(x))=x^2+x-2$
c) $p(q(x))=x-2$ ; $D=[-3,\infty)$

a) $(-1,3)$ and $(4,8)$
b) $x=3$ (the value $x=-2$ is extraneous)
c) $(3,\infty)$

Real World

a) $R(x)=15x-0.025x^2$ ; $P(x)=-0.025x^2+11x-400$
b) Break even at $40$ or $400$ meals; profit for $40<x<400$ (41 to 399 meals).
c) 150 meals at $11.25 each, for a profit of $687.50. The vertex ( $x=220$ , $810) is out of reach, and $P$ is still increasing at $x=150$ .

a) $d(0.5)\approx-8.5$ : 8.5 cm below rest; $d(1)=6.0$ : 6 cm above rest.
b) After about $4.6$ s ( $t>\log_2 24\approx4.58$ ).
c) About $0.012$ cm (0.1 mm). The model never reaches $0$ , since $(0.5)^t>0$ , so it never stops. A real board stops because of friction, and a 0.1 mm swing can't be seen: past about 5 s the model describes no visible motion.

a) $T(h(t))=15-1.95t$ ; input: minutes since launch; output: the air temperature (°C) at the balloon.
b) After $20$ minutes, at $6$ km.
c) $T(h(50))=-82.5$ °C, but $h(50)=15$ km is above 11 km, where $T$ no longer applies (real air there stays near $-56$ °C). Domain: $0\le t\le\frac{110}{3}$ , about $[0,36.7]$ minutes.

Think It Through

a) $x=-1$ : $-1\ge-4$ is true, but the student left it out. $x=-3$ : $-27\ge-12$ is false, but the student included it.
b) Dividing by $x$ : the sign of $x$ is unknown, so for $x<0$ the inequality should flip, and $x=0$ (a solution) is lost.
c) $[-2,0]\cup[2,\infty)$

a) e.g. $g(x)=3x-1$ , $f(x)=x^5+2$
b) e.g. $g(x)=x^2+1$ , $f(x)=\sqrt{x}$ ; or $g(x)=x^2$ , $f(x)=\sqrt{x+1}$
c) e.g. $f(x)=x+2$ , $g(x)=x-5$ : both composites equal $x-3$ . Or $f(x)=2x$ , $g(x)=3x$ : both equal $6x$ .

Boss Level

Boss

a) $f_2(x)=\dfrac{x-1}{x}$ ( $x\ne0,1$ )
b) $f_3(x)=x$ ( $x\ne0,1$ )
c) $-1$ (since $2026=3\cdot675+1$ , $f_{2026}(2)=f(2)$ )

Free cheat sheet

Combining Functions Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

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Most testedFor $f(x)=\sqrt{x-1}$ and $g(x)=x^2+2$ , find $f(g(x))$ and $g(f(x))$ with their domains. (card 4)

1Sums and differences

$(f\pm g)(x)=f(x)\pm g(x)$

UseDomain: where both $f$ and $g$ are defined. Graph: add (or subtract) the $y$ -values at each $x$ .

Example $f(x)=\sqrt x$ , $g(x)=\sqrt{4-x}$ : $f+g$ has domain $[0,4]$ .

TrapSubtract every term of $g$ : $(x^2+1)-(2x-3)=x^2-2x+4$ , not $x^2-2x-2$ .

2Products and quotients

$(fg)(x)=f(x)\,g(x)\qquad\left(\dfrac fg\right)(x)=\dfrac{f(x)}{g(x)}$

UseDomain of $f/g$ : both defined and $g(x)\ne0$ , found before simplifying. Sign of $fg$ : multiply the signs. $y=ae^{-kx}\sin x$ : the amplitude $ae^{-kx}$ shrinks toward $0$ but never reaches it.

Trap $(x^2-9)/(x+3)$ simplifies to $x-3$ but still excludes $x=-3$ .

3Composition

$(f\circ g)(x)=f(g(x))$

UseInside first: apply $g$ , then $f$ . Replace every $x$ in $f$ with $g(x)$ . In general $f(g(x))\ne g(f(x))$ , and $f\circ g$ is not $fg$ .

Example $f(x)=x^2$ , $g(x)=x+3$ : $f(g(x))=(x+3)^2$ , but $g(f(x))=x^2+3$ .

Trap $f(g(2))$ : find $g(2)$ first, then put it into $f$ . Not $f(2)$ combined with $g(2)$ .

4Composite domain and decomposition

UseDomain of $f\circ g$ : $x$ in the domain of $g$ and $g(x)$ in the domain of $f$ . Find it before simplifying. To decompose, name the inner step $g$ and the outer step $f$ .

Example $f(x)=\sqrt{x-1}$ , $g(x)=x^2+2$ : $f(g(x))=\sqrt{x^2+1}$ on $\mathbb{R}$ ; $g(f(x))=x+1$ on $x\ge1$ .

Trap $p(x)=x^2$ , $q(x)=\sqrt{x-2}$ : $p(q(x))=x-2$ , but its domain is $x\ge2$ , not $\mathbb{R}$ .

5Equations and inequalities

Use $f(x)=g(x)$ : solve $f(x)-g(x)=0$ , or read the intersections from a graph. Inequality: everything on one side, factor, test every interval. Squaring both sides can add roots: check them.

TrapA squared factor never changes sign: $(x+1)^2(x-3)<0$ means $x<3$ , $x\ne-1$ .

TrapNever divide by $x$ of unknown sign: $x^2>2x$ is $x<0$ or $x>2$ .

6Modelling

UseProfit = revenue − cost: a difference. Price × quantity: a product. One quantity feeds the next: a composition, inner first. Check the domain against the context.

ExampleFuel $V(d)=0.08d$ L and cost $C(V)=1.6V$ : cost of a trip $C(V(d))=0.128d$ .

TrapUse the vertex only if it is in the allowed domain, such as below a capacity limit.

Which combination?

Quantities add or subtract

f\pm g

One quantity scales another

fg

One output is the next input

f(g(x))

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