1.7 Inverse Functions

Inverse Functions

Inverse functions reverse what a function does. If $f$ takes an input and produces an output, then $f^{-1}$ takes that output and returns the original input. This concept is essential for solving equations where you need to "undo" operations, and it shows up constantly in later topics like logarithms (which are inverses of exponentials) and inverse trig functions.

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Verification of Inverse Functions

Two functions $f$ and $g$ are inverses if and only if both of these are true:

• $f(g(x)) = x$ for all $x$ in the domain of $g$
• $g(f(x)) = x$ for all $x$ in the domain of $f$

Both compositions must simplify to $x$. Checking only one isn't enough.

Example: Verify that $f(x) = 2x + 1$ and $g(x) = \frac{x - 1}{2}$ are inverses.

1. Compute $f(g(x)) = 2\left(\frac{x-1}{2}\right) + 1 = (x - 1) + 1 = x$ ✓

2. Compute $g(f(x)) = \frac{(2x + 1) - 1}{2} = \frac{2x}{2} = x$ ✓

Since both equal $x$, these functions are confirmed inverses.

Evaluating inverses from known values: If you know that $f(3) = 9$, then $f^{-1}(9) = 3$. You're just reading the relationship backwards. This works with tables and graphs too: find the output in the original function, and the corresponding input is the inverse's output.

A quick note on notation: $f^{-1}(x)$ means the inverse function of $f$. It does not mean $\frac{1}{f(x)}$. That's a common mistake worth watching for.

Domains and Ranges of Inverses

For a one-to-one function $f$ with domain $A$ and range $B$, its inverse $f^{-1}$ has domain $B$ and range $A$. The domain and range swap completely.

Why does the function need to be one-to-one? If two different inputs give the same output, the inverse wouldn't know which input to return. For example, $f(x) = x^2$ gives $f(3) = 9$ and $f(-3) = 9$. So what would $f^{-1}(9)$ be? There's no single answer, which means no inverse exists unless you restrict the domain.

Domain restrictions fix this problem by cutting the function down to a portion that is one-to-one:

• $f(x) = x^2$ with $x \geq 0$ becomes one-to-one, and its inverse is $f^{-1}(x) = \sqrt{x}$
• Trig functions like $\sin(x)$ need restricted domains to define $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$

When you restrict the domain of $f$, that restricted domain becomes the range of $f^{-1}$, and vice versa.

Finding Inverse Functions

To find an inverse algebraically:

1. Replace $f(x)$ with $y$, so you have $y = f(x)$
2. Swap $x$ and $y$ to get $x = f(y)$
3. Solve for $y$ in terms of $x$
4. Write the result as $f^{-1}(x) = \ldots$

Example: Find the inverse of $f(x) = 3x - 2$.

1. $y = 3x - 2$

2. Swap: $x = 3y - 2$

3. Solve: $x + 2 = 3y$, so $y = \frac{x + 2}{3}$

4. $f^{-1}(x) = \frac{x + 2}{3}$

You can verify by composing: $f(f^{-1}(x)) = 3\left(\frac{x+2}{3}\right) - 2 = x + 2 - 2 = x$ ✓

Graphically, the inverse is found by reflecting the graph of $f(x)$ over the line $y = x$. For instance, the graph of $f(x) = 2^x$ reflected over $y = x$ gives $f^{-1}(x) = \log_2 x$. This is actually why logarithms and exponentials have the relationship they do.

Graphing Inverses by Reflection

The line $y = x$ is the mirror line between any function and its inverse. The core idea: if $(a, b)$ is on the graph of $f$, then $(b, a)$ is on the graph of $f^{-1}$.

To graph an inverse by reflection:

1. Plot the original function $f(x)$ (or identify key points on it)
2. Draw the line $y = x$ as a reference
3. Swap the coordinates of each point: $(a, b) \rightarrow (b, a)$
4. Connect the reflected points to form the graph of $f^{-1}(x)$

Example: For $f(x) = 2x$, some points are $(0, 0)$, $(1, 2)$, $(2, 4)$. Reflecting gives $(0, 0)$, $(2, 1)$, $(4, 2)$, which lie on $f^{-1}(x) = \frac{x}{2}$. Both graphs are symmetric across $y = x$.

This reflection property also gives you a visual check: if you fold the graph along $y = x$, the function and its inverse should land on top of each other.

Properties of Inverse Functions

• One-to-one requirement: A function must pass the horizontal line test to have an inverse. If any horizontal line crosses the graph more than once, the function is not one-to-one and has no inverse (without a domain restriction).
• Monotonicity: If a function is strictly increasing or strictly decreasing over its entire domain, it's guaranteed to be one-to-one and therefore has an inverse. This is a useful shortcut for determining invertibility.
• Bijection: Formally, a function needs to be both injective (one-to-one) and surjective (onto its stated range) to have an inverse. In practice for this course, the one-to-one condition is what you'll check most often.
• Inverse trig functions ($\arcsin$, $\arccos$, $\arctan$) are the classic examples of using domain restrictions to create invertible functions. You'll work with these extensively in later units.