Secondary 5 Operations on Functions Worksheet

All 6 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Operations on Functions

   Let $f(x)=\sqrt{x-4}$ and $g(x)=x-9$.

   1. Evaluate $(f+g)(8)$ and $(f·g)(13)$.
   2. Give the domain of ${\displaystyle \frac{f}{g}}$ in Québec interval notation, and name the two different restrictions that produce it.
2. Q2The Composition of Functions

   Let $f(x)=2x-5$ and $g(x)=x^2+1$.

   1. Find the rules of $(f\circ g)(x)$ and $(g\circ f)(x)$, and use them to show that composition is not commutative.
   2. Evaluate $(f\circ g)(3)$ and solve $(g\circ f)(x)=1$.
3. Q3The Difference of Functions

   Let $f(x)=\sqrt{x-2}$ and $g(x)=3x-1$.

   1. Write the rule of $(f-g)(x)$ and the rule of $(g-f)(x)$, and give the domain of each.
   2. Evaluate $(f-g)(6)$.
   3. A student claims that $f-g$ and $g-f$ are “the same function written the other way round”. Explain why this is false, then determine whether there is any value of $x$ at which the two rules give the same number.
4. Q4The Product of Functions

   Let $f(x)=\sqrt{x+4}$ and $g(x)={\displaystyle \frac{1}{x-2}}$. Give the rule of $(f·g)(x)$, determine its domain, and evaluate $(f·g)(0)$.

5. Q5The Quotient of Functions

   Let $f(x)=x^2-9$ and $g(x)=x-3$, both defined on $ℝ$.

   1. Give the simplified rule of $\left({\displaystyle \frac{f}{g}}\right)(x)$ and its domain.
   2. Evaluate $\left({\displaystyle \frac{f}{g}}\right)(5)$.
   3. The simplified rule is a first-degree polynomial, which is defined for every real number. Explain why $x=3$ must nevertheless be excluded.
6. Q6The Sum of Functions

   Let $f(x)=2x-5$, defined on $ℝ$, and $g(x)=\sqrt{x-1}$.

   1. Give the rule and the domain of $(f+g)$.
   2. Evaluate $(f+g)(10)$.
   3. Explain why the domain of a sum of two functions is the intersection of their domains, using $x=0$ as an illustration.

The 5 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Every question on this sheet is really a domain question

The arithmetic of combining two functions is the easy half. You add the rules, you multiply the rules, you substitute one rule into the other. The half that carries the marks is stating where the result exists, and that is decided before you simplify anything. Q1, Q3, Q4, Q5 and Q6 all ask for it explicitly.

The quotient: why a cancelled factor still restricts

Q5 is short and it is the question students most often get half-right. You factor, you cancel, and the rule that comes out is a first-degree polynomial — defined for every real number. The question then asks you why one value must nevertheless be excluded.

f − g and g − f are opposites, not the same function

Q3(c) puts a student's claim in front of you: that f − g and g − f are "the same function written the other way round". Subtraction is not commutative, and the precise statement is stronger and more useful:

So the two rules are opposites. Two opposite numbers are equal only when both are zero, which turns the second half of the question into something concrete: the two functions agree exactly where f(x) = g(x). Set the two original rules equal and solve.

When one of them is a radical, that equation is solved by isolating the radical and squaring — and squaring is a one-way street, so every candidate has to be substituted back into the original equation before you keep it. If the resulting quadratic has a negative discriminant there is nothing to check: no real solution means the two functions never take the same value anywhere, which is itself a complete and well-marked answer.

Composition: substitute the whole rule, and mind the order

(f ∘ g)(x) means f(g(x)): you evaluate g first and feed the result into f. Read it right to left. In practice you replace every x in the rule of f by the entire rule of g, brackets included.

Q2 asks for both orders and then asks you to use them to show composition is not commutative. On my own pair, f(x) = 3x + 1 and g(x) = x²:

The second half of Q2 asks you to solve an equation involving a composition. Two routes work. Compose first and solve the resulting equation, or peel the layers: solve g(u) = 1 for the inner value u, then solve f(x) = u. The peeling route is usually faster and it is much harder to make a sign error in. Whichever you use, expand squares properly — (2x − 5)² is 4x² − 20x + 25, never 4x² − 25, and a perfect-square trinomial appearing at the end is a hint that the equation has a double root rather than two.

Evaluating a combination at a number

Q1(a), Q3(b), Q4 and Q6(b) all ask for a single value. There are two honest routes: build the combined rule and substitute, or evaluate each function at the number and then combine the two results. The second is faster and safer under exam pressure — (f · g)(13) is just f(13) times g(13). Check that the number is in the domain first; an evaluation outside the domain is not a wrong number, it is a non-existent one, and saying so is the correct answer.

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Getting the most out of it

Write both domains before you write the combined rule

Cover the question, write dom f and dom g on their own line, and only then start combining. Nearly every mark lost on this sheet comes from a domain decided after the algebra instead of before it.

Say the composition out loud

"f of g of x" — innermost first. Two seconds of saying it prevents the single most common error on Q2, which is composing in the order the symbols happen to be written rather than the order they are applied.

Redo Q3 and Q5 with the two functions swapped

Same questions, letters exchanged. The domains move, the sign of the difference flips, and the quotient loses a different value — which is precisely the point both questions are making, felt rather than memorised.

Common questions