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Inverse Functions Reference cheat sheet - grade 8-10

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Math Grade 8-10

Inverse Functions Reference Cheat Sheet

A printable reference covering inverse functions, function notation, domain and range swaps, composition checks, and graph reflection for grades 8-10.

TopicsFunctionsGraphsAlgebraMathematicsStudy Guide
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Inverse functions undo the action of a function and help students connect equations, tables, graphs, and real-world relationships. This reference explains what an inverse means, how to find one, and how to check whether it works. Students need these ideas for algebra, graphing, transformations, and later work with exponential and logarithmic functions.

The core idea is that if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a. Inverses switch inputs and outputs, so the domain of ff becomes the range of f−1f^{-1}, and the range of ff becomes the domain of f−1f^{-1}. A correct inverse passes the composition tests f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x, and its graph is a reflection across the line y=xy = x.

Key Facts

  • If f(a)=bf(a) = b, then the inverse function satisfies f−1(b)=af^{-1}(b) = a.
  • To find an inverse from an equation, replace f(x)f(x) with yy, switch xx and yy, then solve for yy.
  • The notation f−1(x)f^{-1}(x) means the inverse of f(x)f(x), not the reciprocal 1f(x)\frac{1}{f(x)}.
  • A function has an inverse function only if it is one-to-one, meaning each output comes from exactly one input.
  • The horizontal line test checks whether a graph has an inverse function, since no horizontal line may cross the graph more than once.
  • The domain of ff is the range of f−1f^{-1}, and the range of ff is the domain of f−1f^{-1}.
  • A function and its inverse cancel by composition: f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x.
  • The graphs of f(x)f(x) and f−1(x)f^{-1}(x) are mirror images across the line y=xy = x.

Vocabulary

Inverse function
An inverse function reverses the inputs and outputs of another function, so f−1(b)=af^{-1}(b) = a when f(a)=bf(a) = b.
One-to-one function
A one-to-one function assigns each output to exactly one input, which allows the function to have an inverse function.
Domain
The domain is the set of allowed input values, or xx-values, for a function.
Range
The range is the set of possible output values, or yy-values, for a function.
Composition
Composition means putting one function inside another, such as f(g(x))f(g(x)).
Horizontal line test
The horizontal line test says a function is one-to-one if every horizontal line crosses its graph at most once.

Common Mistakes to Avoid

  • Reading f−1(x)f^{-1}(x) as 1f(x)\frac{1}{f(x)} is wrong because the −1-1 means inverse function, not reciprocal.
  • Forgetting to switch xx and yy is wrong because inverses reverse inputs and outputs before solving for the new output.
  • Assuming every function has an inverse function is wrong because only one-to-one functions have inverses that are also functions.
  • Ignoring domain restrictions is wrong because a function such as f(x)=x2f(x) = x^2 needs a restricted domain, like x≥0x \ge 0, before its inverse is a function.
  • Checking only one composition is incomplete because a full inverse check uses both f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x when both are defined.

Practice Questions

  1. 1 Find the inverse of f(x)=3x−7f(x) = 3x - 7.
  2. 2 Find the inverse of g(x)=x+42g(x) = \frac{x + 4}{2}.
  3. 3 Verify by composition that f(x)=x−5f(x) = x - 5 and f−1(x)=x+5f^{-1}(x) = x + 5 are inverse functions.
  4. 4 Explain why h(x)=x2h(x) = x^2 on all real numbers does not have an inverse function unless its domain is restricted.