Combine and compose functions, including their domains

Alberta Grade 12 Mathematics

Relations and Functions

Combine functions without losing the domain

Functions can be added, multiplied, divided, or put inside one another. In Mathematics 30-1, a correct formula is only part of the answer: the input must also work in every original function and, for a quotient, must not make the denominator zero. A composition has an extra check because the output of the inner function becomes the input of the outer one.

What you will learn

Calculate operations and compositions from two given functions.
State the domain of a quotient and a composition.
Check an answer by evaluating the functions separately.

Operations use the same input

For functions f and g, addition and multiplication combine their outputs at the same x. Division does too, but g(x) must not be zero. The allowed inputs are the intersection of the original domains, with denominator zeros removed for a quotient.

A square root such as

\sqrt{x-1}

requires

x\geq 1

. A rational expression such as

1/(x-2)

requires

x\neq 2

. State restrictions before simplifying: cancellation in an expression does not restore an excluded input.

(f+g)(x)=f(x)+g(x), (

\frac{f}{g}

)(x)=

\frac{f(x)}{g(x)}

, g(x)\ne 0

Composition uses an output as an input

In

(f\circ g)(x)

, calculate g(x) first, then feed that result into f. Reversing the order usually gives a different function. To determine its domain, check that x is allowed in g and that g(x) is allowed as an input to f.

This order is easiest to see with a small substitution table: choose a permitted x, evaluate g(x), and then evaluate f at that result. Compare the numerical output with the composed formula as a check.

(f° g)(x)=f(g(x))

Worked example

One pair of functions, two different domains

Let

f(x)=\sqrt{x-1}

and

g(x)=x-3

. Find

(f/g)(x)

and

(f\circ g)(x)

with their domains.

Function and composition: different starting inputs

The graph window shows f(x) = √(x − 1) beginning at x = 1 and (f ∘ g)(x) = √(x − 4) beginning at x = 4. The quotient is not plotted; its separate exclusion x = 3 still applies.

Start with the original restrictions
The square root needs x at least 1. The linear function accepts every real number.
Divide the outputs
Exclude x = 3 as well because it makes the denominator zero.
( $\frac{f}{g}$ )(x)= $\frac{\sqrt{x-1}}{x-3}$ , x≥ 1,\ x\ne 3
Compose in the stated order
First calculate g(x) = x − 3. The outer square root then needs x − 3 at least 1.
(f° g)(x)=f(x-3)= $\sqrt{x-4}$ , x≥ 4

Answer: The quotient is

\sqrt{x-1}/(x-3)

for

x\geq1

,

x\neq3

. The composition is

\sqrt{x-4}

for

x\geq4

.

Check: At x = 5, g(5) = 2 and f(2) = 1, agreeing with

\sqrt{5-4}=1

.

Common mistakes and how to avoid them

Using the domain of f alone for a composition.

Correction: The inner output must lie inside the outer function’s domain; solve that condition after substitution.

Cancelling a factor and forgetting an original denominator restriction.

Correction: Record excluded inputs from the original expression before simplifying.

Lesson summary

Operations combine outputs at the same x.
Composition evaluates the inner function first.
Every domain restriction must be checked against the original functions.

Check your understanding

Question 1

If

f(x)=\sqrt{x}

and

g(x)=x+2

, what is the domain of

(f\circ g)(x)

?

$x\geq -2$
$x\geq 0$
All real numbers

Show answer and explanation

x\geq -2

The outer square root receives x + 2, so x + 2 must be nonnegative.

Key terms

Domain: The set of input values for which a function is defined.
Composition: A function made by using one function’s output as another function’s input.

Continue through Math 30-1

View the complete Math 30-1 Alberta Grade 12 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation RF1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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RF1 · Combine and compose functions, including their domains