4.6 Inverse Functions & Their Properties

4.6 Inverse Functions & Their Properties

In mathematics, an inverse function undoes or reverses the operational effect of the original function. If a function $f$ maps an input $x$ to an output $y$, its inverse function $f^{-1}$ maps that output $y$ directly back to the original input $x$.

$$f(x) = y \iff f^{-1}(y) = x$$

NOTATION WARNING: The superscript $-1$ in $f^{-1}(x)$ denotes an inverse function, NOT an exponent or reciprocal! $f^{-1}(x) \ne \frac{1}{f(x)}$. The multiplicative inverse (reciprocal) is written as $[f(x)]^{-1} = \frac{1}{f(x)}$.

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1. One-to-One Functions and the Horizontal Line Test

Not every function possesses a valid inverse function. For an inverse relation to satisfy the Vertical Line Test (and thus be a true function), the original function must be one-to-one (1-to-1).

• Formal Definition: A function $f$ is one-to-one if no two distinct inputs $x_1 \ne x_2$ map to the exact same output value $f(x_1) = f(x_2)$. Algebraically: $$f(x_1) = f(x_2) \implies x_1 = x_2$$

The Horizontal Line Test (HLT)

Graphically, a function $f(x)$ has a valid inverse function $f^{-1}(x)$ if and only if no horizontal line intersects the graph of $f(x)$ in more than one point.

• Fails HLT: The parabola $f(x) = x^2$ fails the horizontal line test (e.g., horizontal line $y = 4$ intersects at $(-2, 4)$ and $(2, 4)$). Reversing these points gives $(4, -2)$ and $(4, 2)$, which fails the Vertical Line Test. Thus, $f(x) = x^2$ is not one-to-one on $(-\infty, \infty)$ and lacks a general inverse function.
• Passes HLT: The cubic curve $f(x) = x^3$ passes the horizontal line test everywhere. It is one-to-one and possesses a valid inverse function $f^{-1}(x) = \sqrt[3]{x}$.

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2. Fundamental Inverse Cancellation Identities & Domain-Range Swapping

If $f$ and $f^{-1}$ are inverse functions, they satisfy two mandatory cancellation equations:

1. $f\Big(f^{-1}(x)\Big) = x \quad \text{for all } x \in \text{Domain}\left(f^{-1}\right)$
2. $f^{-1}\Big(f(x)\Big) = x \quad \text{for all } x \in \text{Domain}(f)$

Complete Domain and Range Interchange

Because an inverse function swaps input and output roles, the domain and range of $f$ and $f^{-1}$ are completely interchanged:

$$\text{Domain}\left(f^{-1}\right) = \text{Range}(f)$$ $$\text{Range}\left(f^{-1}\right) = \text{Domain}(f)$$

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3. Four-Step Algebraic Algorithm to Compute $f^{-1}(x)$

To derive the explicit algebraic equation for the inverse of a one-to-one function $f(x)$:

1. Step 1: Replace $f(x)$ with $y$.
2. Step 2: Interchange the variables $x$ and $y$ (replace every $x$ with $y$, and $y$ with $x$).
3. Step 3: Solve the resulting equation for $y$ in terms of $x$.
4. Step 4: Replace $y$ with inverse notation $f^{-1}(x)$, and state any domain restrictions.

Worked Example 1: Rational Function Inverse

Find the inverse of $f(x) = \frac{4x + 1}{x - 3}$ for $x \ne 3$.

1. Step 1 (Set $y$): $$y = \frac{4x + 1}{x - 3}$$
2. Step 2 (Swap $x$ and $y$): $$x = \frac{4y + 1}{y - 3}$$
3. Step 3 (Solve for $y$): Cross-multiply: $x(y - 3) = 4y + 1 \implies xy - 3x = 4y + 1$. Collect $y$ terms on one side: $$xy - 4y = 3x + 1 \implies y(x - 4) = 3x + 1 \implies y = \frac{3x + 1}{x - 4}$$
4. Step 4 (Write $f^{-1}(x)$): $$f^{-1}(x) = \frac{3x + 1}{x - 4}, \quad x \ne 4$$

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4. Restricting Domains of Non-One-to-One Functions

When a function fails the Horizontal Line Test over its natural domain, we can restrict its domain to a monotonic sub-interval where it passes the HLT, allowing an inverse function to be constructed.

Worked Example 2: Domain-Restricted Quadratic Inverse

Find the inverse of $f(x) = (x - 2)^2 + 5$ restricted to domain $x \ge 2$.

1. Domain & Range Analysis:
   • Restricted Domain of $f$: $[2, \infty)$.
   • Range of $f$: Since $(x-2)^2 \ge 0$, $f(x) \ge 5 \implies [5, \infty)$.
2. Swap and Solve: $$x = (y - 2)^2 + 5 \implies x - 5 = (y - 2)^2$$ Take the principal (positive) square root because $y \ge 2$: $$\sqrt{x - 5} = y - 2 \implies y = \sqrt{x - 5} + 2$$
3. Conclusion: $$f^{-1}(x) = \sqrt{x - 5} + 2 \quad \text{with Domain } [5, \infty)$$

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5. Geometric Reflection Property Across $y = x$

The graph of $y = f^{-1}(x)$ is a precise reflection of the graph of $y = f(x)$ across the identity line $y = x$.

• If ordered pair $(a, b)$ lies on the graph of $f(x)$, then the reversed ordered pair $(b, a)$ must lie on the graph of $f^{-1}(x)$.
• Any points of intersection between $f(x)$ and $f^{-1}(x)$ must lie directly along the line of symmetry $y = x$.