Functions, One-to-One Functions, and Inverse Functions: Key Concepts and Applications

Functions and Their Properties

Vertical Line Test

The Vertical Line Test is a graphical method used to determine whether a relation is a function. A relation is a function if and only if no vertical line intersects its graph at more than one point.

• Definition: A relation is a function if every input (x-value) corresponds to exactly one output (y-value).

• How to use: Draw vertical lines through the graph. If any vertical line crosses the graph more than once, the relation is not a function.

• Purpose: To verify if a graph represents a function.

• Example: The graph of a parabola opening sideways fails the vertical line test and is not a function.

Horizontal Line Test

The Horizontal Line Test is used to determine if a function is one-to-one (1-1). A function is one-to-one if and only if no horizontal line intersects its graph at more than one point.

• Definition: A function is one-to-one if each output (y-value) is paired with exactly one input (x-value).

• How to use: Draw horizontal lines through the graph. If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Purpose: To determine if a function has an inverse that is also a function.

• Example: The graph of $f(x) = x^2$ fails the horizontal line test, so it is not one-to-one.

Key Values in Functions

• If a relation is a function: The x-values (inputs) cannot repeat.

• If a function is one-to-one: The y-values (outputs) cannot repeat.

Determining Functions and One-to-One Correspondence

Given sets of ordered pairs, determine if the relation is a function and if it is one-to-one.

• Function: No x-value repeats.

• One-to-one: No y-value repeats.

Set	Function?	One-to-One?
{(7,7), (8,8), (9,45), (10,65)}	Yes	Yes
{(2,9), (5,9), (2,14), (6,9)}	No (x=2 repeats)	No (y=9 repeats)

Inverse Functions

Definition and Notation

An inverse function $f^{-1}(x)$ reverses the effect of the original function $f(x)$. For a function to have an inverse that is also a function, it must be one-to-one.

• Notation: $f^{-1}(x)$ denotes the inverse of $f(x)$.

• Existence: Only one-to-one functions have inverses that are also functions.

Steps to Find the Inverse of a Function

1. Set the function equal to $y$.

2. Switch $x$ and $y$ in the equation.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$.

Example: Find the inverse of $f(x) = 3x + 1$.

• Set $y = 3x + 1$

• Switch $x$ and $y$: $x = 3y + 1$

• Solve for $y$: $y = \frac{x - 1}{3}$

• So, $f^{-1}(x) = \frac{x - 1}{3}$

Graphical Interpretation of Inverses

• Taking the inverse of a function reflects its graph over the line $y = x$.

• If $(a, b)$ is on the graph of $f(x)$, then $(b, a)$ is on the graph of $f^{-1}(x)$.

Domain and Range of Inverse Functions

• The domain of $f(x)$ becomes the range of $f^{-1}(x)$, and vice versa.

• Notation:

  • $D_f$ = domain of $f(x)$

  • $R_f$ = range of $f(x)$

  • $D_{f^{-1}}$ = domain of $f^{-1}(x)$

  • $R_{f^{-1}}$ = range of $f^{-1}(x)$

Examples: Finding Inverses

• $f(x) = x + 4$ $y = x + 4$ $x = y + 4$ $y = x - 4$ $f^{-1}(x) = x - 4$

• $f(x) = 3x + 1$ $y = 3x + 1$ $x = 3y + 1$ $x - 1 = 3y$ $y = \frac{x - 1}{3}$ $f^{-1}(x) = \frac{x - 1}{3}$

• $f(x) = x^3 + 4$ $y = x^3 + 4$ $x = y^3 + 4$ $x - 4 = y^3$ $y = \sqrt[3]{x - 4}$ $f^{-1}(x) = \sqrt[3]{x - 4}$

• $f(x) = (x + 9)^3$ $y = (x + 9)^3$ $x = (y + 9)^3$ $\sqrt[3]{x} = y + 9$ $y = \sqrt[3]{x} - 9$ $f^{-1}(x) = \sqrt[3]{x} - 9$

• $f(x) = x^3 + 8$ $y = x^3 + 8$ $x = y^3 + 8$ $x - 8 = y^3$ $y = \sqrt[3]{x - 8}$ $f^{-1}(x) = \sqrt[3]{x - 8}$

• $f(x) = 4x + 1$ $y = 4x + 1$ $x = 4y + 1$ $x - 1 = 4y$ $y = \frac{x - 1}{4}$ $f^{-1}(x) = \frac{x - 1}{4}$

Graphical Examples

• Given a point $(a, b)$ on the graph of a one-to-one function, the point $(b, a)$ will be on the graph of its inverse.

• Graphing both $f(x)$ and $f^{-1}(x)$ will show symmetry over the line $y = x$.

Composition of Functions and Inverses

If two functions $f(x)$ and $g(x)$ are inverses of each other, then:

• $f(g(x)) = x$

• $g(f(x)) = x$

This property can be used to verify if two functions are inverses.

Examples: Verifying Inverses

• $f(x) = \frac{3}{x - 4}$ and $g(x) = \frac{3}{x} + 4$

• $f(x) = -2x$ and $g(x) = -2x$

• To verify, compute $f(g(x))$ and $g(f(x))$ and check if both simplify to $x$.

Summary Table: Function and Inverse Properties

Property	Function $f(x)$	Inverse $f^{-1}(x)$
Domain	$D_f$	$R_f$
Range	$R_f$	$D_f$
Graphical Test	Vertical Line Test	Horizontal Line Test (for 1-1)
Point Mapping	$(a, b)$	$(b, a)$

Key Takeaways

• The vertical line test determines if a relation is a function.

• The horizontal line test determines if a function is one-to-one (and thus invertible).

• To find the inverse of a function, switch $x$ and $y$ and solve for $y$.

• The domain of a function becomes the range of its inverse, and vice versa.

• Two functions are inverses if their compositions yield the identity function: $f(g(x)) = x$ and $g(f(x)) = x$.