Finding Inverse Functions

Algebra 2 · Axiom Academy

LESSON Finding Inverse Functions Learn how to find inverse functions algebraically and graphically, understand the reflection property, and use the horizontal line test. 1. What is an Inverse Function? The Reflection Property: The graph of f -1 is the reflection of the graph of f over the line y = x . This means if point (a, b) is on f , then point (b, a) is on f -1 . Not every function has an inverse. A function must be one-to-one to have an inverse—this means each output corresponds to exactly one input. Each horizontal line hits the graph once Some horizontal lines hit twice So is the square root the inverse of the square? Only after you restrict the domain . On all of , f(x) = x^2 fails the test and has no inverse. But f(x) = x^2 restricted to is one-to-one, and that function's inverse is . The restriction is not a technicality. Forget it and you lose roots: from x^2 = 9 you would "invert" to x = 3 and miss x = -3 . The graphical relationship between a function and its inverse is beautiful: they are mirror images across the line y = x. Why? If (a, b) is on f(x), then f(a) = b. For the inverse, we want f -1 (b) = a, which means (b, a) is on f -1 (x). Swapping coordinates creates a reflection over y = x. If (3, 7) is on f(x), then (7, 3) is on f -1 (x) The domain of f becomes the range of f -1 The range of f becomes the domain of f -1 4. Finding Inverses Algebraically To find the inverse of a function algebraically, follow these steps:

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