Inverse Functions and One-to-One Functions
Master the concepts of function invertibility, the horizontal line test, and finding inverses algebraically.
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Questions Covered in This Set
10 cards to master
What is a one-to-one function?
A function where each output corresponds to exactly one input; different inputs must produce different outputs.
What is the horizontal line test?
If any horizontal line intersects a graph more than once, the function is not one-to-one and doesn't have an inverse.
What are the four steps to find an inverse algebraically?
1) Replace f(x) with y, 2) Swap x and y, 3) Solve for y, 4) Replace y with f⁻¹(x).
What is the graphical relationship between a function and its inverse?
They are reflections across the line y = x.
What are the two composition properties of inverse functions?
f(f⁻¹(x)) = x for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f.
Why doesn't f(x) = x² have an inverse over all real numbers?
Because it's not one-to-one; both f(2) = 4 and f(-2) = 4, so two different inputs give the same output.
How can you make a non-one-to-one function invertible?
Restrict the domain to create a one-to-one function on that restricted domain.
If point (3, 7) is on the graph of f, what point is on the graph of f⁻¹?
The point (7, 3) is on the graph of f⁻¹.
What happens to domain and range when finding an inverse?
The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹.
Does f(x) = x³ pass the horizontal line test?
Yes, every horizontal line crosses f(x) = x³ exactly once, so it is one-to-one and invertible.
