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MathidiaMathematics Academy
Functions and graph transformations
Lesson 6 of 75
Introduction

Functions, domains and ranges

Objective. Use function notation and determine valid domains and ranges.

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  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A function f(x) assigns exactly one output to each input in its domain, and the range is the resulting set of possible outputs. Domains must exclude values causing division by zero or negative numbers under an even root, such as f(x)=1/(x−3) needing x≠3. To find the range, consider the graph's shape: for f(x)=x²+1 with domain all real x, every output is at least 1, so the range is f(x)≥1. Always state domain and range using correct interval or inequality notation, matching the function's actual behaviour rather than guessing.

Worked example

State the largest possible domain of the function f(x) = 1/√(x−5).

1 … 5
Illustrated card showing 1 … 5
  1. 1Note the denominator cannot be zero or negative, since it is a square root
  2. 2Require x−5>0 strictly, not just ≥0
  3. 3Solve the strict inequality to get x>5
  4. 4State the domain as x>5

Answer: x > 5

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Common mistake. Students assume the range always matches the domain's shape, claiming f(x)=√x has range all real numbers; since square roots give non-negative outputs only, the correct range is f(x)≥0.

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