Inverse Functions
Undo a function; graph reflects across y = x.
Key idea
Undo a function; graph reflects across y = x.
In depth
An inverse f⁻¹ undoes f, so f⁻¹(f(x)) = x. To find it, swap x and y and solve for y. Graphs are mirror images across y = x. Only one-to-one functions have inverses.
Worked example
f(x) = 3x − 2 → x = 3y − 2 → y = (x + 2)/3, so f⁻¹(x) = (x + 2)/3.
Formula
Real-world uses
- Converting Celsius to Fahrenheit and back
- Decoding a message or reversing a calculation
Common mistake
Reading f⁻¹(x) as 1/f(x). It is not the reciprocal.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Inverse Functions?
Undo a function; graph reflects across y = x. It is a topic at the advanced level in Algebra.
What is the formula for Inverse Functions?
The key formula is f(f⁻¹(x)) = x.
Where is Inverse Functions used in real life?
Real-world examples: Converting Celsius to Fahrenheit and back; Decoding a message or reversing a calculation.
What should I learn before Inverse Functions?
Make sure you are comfortable with Function Composition, the previous topic in Algebra.
What is a common mistake with Inverse Functions?
Reading f⁻¹(x) as 1/f(x). It is not the reciprocal.
How can I practise Inverse Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Inverse Functions?
Move on to Even and Odd Functions next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.