Precalculus Inverse Functions Lesson (Free PDF)

Precalculus math lesson: Inverse Functions – Maths PDF

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This inverse functions lesson for Precalculus covers inverse functions (Common Core F-BF.B.4a, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d), with a free PDF to print.

An inverse function f^-1 undoes f: if f(a) = b, then f^-1(b) = a. Only one-to-one functions, which pass the horizontal line test, have inverses. To find an inverse, write y = f(x), swap x and y, and solve for y. The graph of f^-1 is the reflection of f over the line y = x.

Precalculus · Common Core State Standards: F-BF.B.4a, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d

Key facts in this inverse functions lesson

Key facts

f^-1(f(x)) = x and f(f^-1(x)) = x.
Find an inverse: write y = f(x), swap x and y, solve for y.
Domain of f becomes the range of f^-1, and the range of f becomes the domain of f^-1.
The graph of f^-1 is the reflection of the graph of f over y = x.
Restrict the domain of a function that is not one-to-one, such as x^2 with x >= 0.

What is an inverse function?

The inverse of \(f\), written \(f^{-1}\), undoes \(f\). If \(f(a)=b\), then \(f^{-1}(b)=a\). That means \(f^{-1}(f(x))=x\) for every \(x\) in the domain of \(f\), and \(f(f^{-1}(x))=x\) for every \(x\) in the domain of \(f^{-1}\). The notation \(f^{-1}\) is not a reciprocal: \(f^{-1}(x)\neq\frac{1}{f(x)}\).

One-to-one

A function is one-to-one if each output comes from exactly one input. Only one-to-one functions have inverse functions. The horizontal line test: no horizontal line meets the graph more than once.

A parabola fails the horizontal line test: y = 4 hits it twice.

How do you find an inverse algebraically?

Four steps for finding f inverse.

Example: \(f(x)=2x+1\). Write \(y=2x+1\). Swap: \(x=2y+1\). Solve: \(y=\frac{x-1}{2}\). So \(f^{-1}(x)=\frac{x-1}{2}\). Check: \(f(f^{-1}(x))=2\cdot\frac{x-1}{2}+1=x\).

Why does this work? The rule \(f\) multiplies by 2 and then adds 1. To undo it, reverse the steps in the opposite order: subtract 1, then divide by 2. Swapping \(x\) and \(y\) and solving does exactly that automatically, even for rules with several steps. When the rule has a square, a root or a fraction, check the domain at the end, because the inverse may be valid only for some inputs.

How are the graphs of f and its inverse related?

Swapping \(x\) and \(y\) reflects the graph over the line \(y=x\). Every point \((a,b)\) on \(f\) becomes \((b,a)\) on \(f^{-1}\). The domain of \(f\) becomes the range of \(f^{-1}\), and the range of \(f\) becomes the domain of \(f^{-1}\).

The point (1, 3) on f matches (3, 1) on the inverse.

What if the function is not one-to-one?

Restrict the domain to a piece that passes the horizontal line test. The function \(f(x)=x^{2}\) is not one-to-one, but \(f(x)=x^{2}\) with \(x\ge 0\) is. Its inverse is \(f^{-1}(x)=\sqrt{x}\), with domain \(x\ge 0\).

With x at least 0, the square and the square root are inverses.

For a square-root function such as \(f(x)=\sqrt{x-2}\) the inverse is \(f^{-1}(x)=x^{2}+2\), but only for \(x\ge 0\), because outputs of \(f\) are never negative.

Inverse functions lesson: questions and answers

Is f^-1(x) the same as 1/f(x)?

No. The exponent -1 here means inverse function, not reciprocal. For f(x) = 2x + 1, f^-1(x) = (x – 1)/2, while 1/f(x) = 1/(2x + 1).

How do you check that two functions are inverses?

Compute both compositions. If f(g(x)) = x and g(f(x)) = x, the functions are inverses of each other.

Why does y = x^2 need a restricted domain?

Because 2 and -2 both give 4, so one output has two inputs. Restricting to x >= 0 makes it one-to-one, so the inverse is the square root.

What is the horizontal line test?

If every horizontal line crosses the graph at most once, the function is one-to-one and has an inverse. If some horizontal line crosses twice, it has no inverse on that domain.

How do the domain and range of an inverse relate to f?

They swap places. The domain of f^-1 equals the range of f, and the range of f^-1 equals the domain of f.

How to use this inverse functions 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.

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