Inverse functions ($f^{-1}(x)$)

1. Definition of one-to-one functions and the horizontal line test

One-to-One Functions and the Horizontal Line Test

A function $f$ is one-to-one (injective) if each output value corresponds to exactly one input value. Equivalently, if $f(a) = f(b)$, then $a = b$ must hold.

The horizontal line test provides a visual criterion: a function is one-to-one if and only if every horizontal line intersects its graph at most once.

Core Rules:

• A function must be one-to-one to have an inverse function.
• If any horizontal line crosses the graph more than once, the function is not one-to-one.
• Strictly increasing or strictly decreasing functions are always one-to-one.
• Functions like $f(x) = x^2$ (on all real numbers) fail the test because $f(2) = f(-2) = 4$.

Only one-to-one functions guarantee that the inverse relation is also a function.

Example: $f(x) = 2x + 3$ passes the horizontal line test (strictly increasing), so it has an inverse. But $g(x) = x^2$ does not pass (e.g., the line $y = 4$ intersects at $x = 2$ and $x = -2$).

2. Algebraic steps to find the inverse: swapping $x$ and $y$

Algebraic Steps to Find the Inverse

To find the inverse function $f^{-1}(x)$ algebraically, we reverse the input-output relationship of $f(x)$. The standard procedure involves swapping variables and solving.

Core Rules:

• Step 1: Replace $f(x)$ with $y$: write $y = f(x)$.
• Step 2: Swap $x$ and $y$ to reverse the roles: write $x = f(y)$.
• Step 3: Solve the resulting equation for $y$ in terms of $x$.
• Step 4: Replace $y$ with $f^{-1}(x)$ to denote the inverse function.

This process works only if $f$ is one-to-one. The swapping step reflects the idea that the inverse undoes the original function.

Example: For $f(x) = 2x + 3$, write $y = 2x + 3$. Swap: $x = 2y + 3$. Solve: $2y = x - 3$, so $y = \frac{x - 3}{2}$. Thus $f^{-1}(x) = \frac{x - 3}{2}$.

3. Graphical relationship between a function and its inverse (symmetry across $y=x$)

Graphical Symmetry Across $y = x$

The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y = x$. This symmetry arises because the inverse swaps the roles of inputs and outputs.

If the point $(a, b)$ lies on the graph of $f$, then the point $(b, a)$ lies on the graph of $f^{-1}$. Reflecting across $y = x$ exchanges $x$ and $y$ coordinates.

Core Rules:

• Every point $(a, b)$ on $f$ corresponds to $(b, a)$ on $f^{-1}$.
• The line $y = x$ acts as the mirror axis.
• If $f$ and $f^{-1}$ are graphed together, they are symmetric about $y = x$.
• The domain of $f$ becomes the range of $f^{-1}$, and vice versa.

This geometric relationship provides a quick visual check for inverse correctness.

Example: For $f(x) = 2x$, the point $(1, 2)$ is on $f$. Its inverse $f^{-1}(x) = \frac{x}{2}$ contains $(2, 1)$, the reflection of $(1, 2)$ across $y = x$.

4. Verifying inverse properties using composition: $f(f^{-1}(x)) = x$

Verifying Inverse Properties Using Composition

Two functions $f$ and $g$ are inverses if and only if their compositions yield the identity function: $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the appropriate domains.

This composition test confirms that $f$ and $f^{-1}$ undo each other. Both directions must hold.

Core Rules:

• Forward composition: $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$.
• Backward composition: $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.
• If either composition fails, the functions are not inverses.
• Verification requires checking both compositions, not just one.

This algebraic criterion is the definitive test for inverse relationships.

Example: For $f(x) = 2x + 3$ and $f^{-1}(x) = \frac{x - 3}{2}$, check $f(f^{-1}(x)) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x$. Similarly, $f^{-1}(f(x)) = \frac{(2x + 3) - 3}{2} = x$.

5. Applications: Decoding encrypted data or finding necessary inputs for a desired output in engineering

Applications of Inverse Functions

Inverse functions solve the problem of reversing a process: given an output, determine the required input. This principle underlies cryptography, engineering design, and data analysis.

In cryptography, encryption functions transform plaintext into ciphertext; the inverse (decryption) recovers the original message. In engineering, if a model predicts output $y = f(x)$ from input $x$, the inverse $f^{-1}(y)$ determines what input produces a target output.

Core Rules:

• Encryption/decryption pairs are inverse functions.
• Inverse functions enable backward reasoning from effects to causes.
• In control systems, inverses calculate required settings to achieve desired performance.
• Temperature conversions (e.g., Celsius to Fahrenheit and back) use inverse relationships.

Inverse functions transform "what happens if" questions into "what is needed for" solutions.

Example: If a temperature conversion is $F = \frac{9}{5}C + 32$, the inverse $C = \frac{5}{9}(F - 32)$ finds Celsius from Fahrenheit. For $F = 77$, we get $C = \frac{5}{9}(77 - 32) = 25$.