Algebra 2 Composition and Inverse Functions Lesson (Free PDF)
This composition inverses lesson for Algebra 2 covers composition and inverse functions (Common Core F-BF.A.1c, F-BF.B.4a), with a free PDF to print.
The composition f(g(x)) applies g first and then f. The inverse of a function undoes it: swap x and y and solve for y, then check that f(f^(-1)(x)) = x. The graph of the inverse is the reflection of the graph across the line y = x. Functions that are not one-to-one need a restricted domain.
Algebra 2 · Common Core State Standards: F-BF.A.1c, F-BF.B.4a
Key facts in this composition inverses lesson
- f(g(x)): apply g first, then f.
- Composition order matters.
- Inverse: swap x and y, then solve for y.
- f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.
- Graphs of f and f^(-1) are mirror images over y = x.
What is the composition of two functions?
To compose functions, use the output of one as the input of the other. The notation \(f(g(x))\) means: apply \(g\) first, then \(f\). For \(f(x) = 2x + 3\) and \(g(x) = x^{2}\), \(f(g(x)) = 2x^{2} + 3\) but \(g(f(x)) = (2x + 3)^{2}\). Order matters.
The domain of \(f(g(x))\) includes only inputs that are in the domain of \(g\) and whose outputs are in the domain of \(f\). For \(f(x) = \frac{1}{x – 2}\) and \(g(x) = x + 3\), \(f(g(x)) = \frac{1}{x + 1}\), so \(x \ne -1\).
What is an inverse function?
Example: \(f(x) = 3x – 6\). Write \(y = 3x – 6\), swap to \(x = 3y – 6\), and solve: \(y = \frac{x + 6}{3}\). So \(f^{-1}(x) = \frac{x + 6}{3}\). Check: \(f\left(\frac{x + 6}{3}\right) = (x + 6) – 6 = x\).
How do you find an inverse when the function is not one-to-one?
The function \(f(x) = x^{2}\) fails the horizontal line test, because \(f(2) = f(-2)\). Restrict the domain to \(x \ge 0\). Then \(f\) is one-to-one and \(f^{-1}(x) = \sqrt{x}\). The restriction decides which root to use: for \(x \le 0\) the inverse would be \(-\sqrt{x}\).
You can also find inverses from tables by swapping the columns, and check any pair by composing both ways. Real-world inverses answer the reverse question: if \(F = \frac{9}{5}C + 32\) converts Celsius to Fahrenheit, the inverse \(C = \frac{5}{9}(F – 32)\) converts back.
Composition inverses lesson: questions and answers
What does f(g(x)) mean?
It means apply g to x first, then apply f to the result. The inner function g works first.
How do you find the inverse of a function?
Write y = f(x), swap x and y, solve for y, and call the result f inverse of x. Then verify that composing in both orders gives x.
How can you tell if two functions are inverses?
Compose them in both orders. If f(g(x)) = x and g(f(x)) = x for all x in the domains, they are inverses.
What does the graph of an inverse look like?
The graph of f inverse is the reflection of the graph of f across the line y = x, because every point (a, b) becomes (b, a).
Why do some functions need a restricted domain to have an inverse?
A function must be one-to-one to have an inverse. For x squared, two inputs give the same output, so we restrict the domain to x greater than or equal to 0.
How to use this composition inverses lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the composition inverses lesson as a PDF with the button below and keep it with your notes.
All Algebra 2 chapters: Algebra 2 math.
More Algebra 2 math
Lessons
See all
Worksheets
See all
Tests
See all
Composition and Inverse Functions Test with Point Values- Radical and Rational Equations Test with Point Values
- Radicals and Rational Exponents Test with Point Values
- Exponential Growth and Decay Test with Point Values
- Arithmetic and Geometric Sequences and Series Test with Point Values
- Sampling, Surveys and Experiments Test with Point Values
