Functions, One-to-One Functions, and Inverse Functions in College Algebra

Functions: Domain, Range, and Composition

Determining the Domain of a Function

The domain of a function is the set of all possible input values (typically x-values) for which the function is defined.

• From the equation: To find the domain, identify values that would make the function undefined (such as division by zero or taking the square root of a negative number).

• From the graph: The domain consists of all x-values for which the graph exists.

• Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x = 2$.

Determining the Range of a Function

The range of a function is the set of all possible output values (y-values).

• Analyze the graph or solve for y in terms of x to determine possible outputs.

• Example: For $f(x) = x^2$, the range is $[0, \infty)$.

Piecewise-Defined Functions

A piecewise-defined function is defined by different expressions for different intervals of the domain.

• Evaluate the function by determining which interval the input belongs to, then use the corresponding expression.

• Example: $f(x) = \begin{cases} 2x+4 & \text{if } x \leq -1 \\ 2x-6 & \text{if } x \geq 4 \end{cases}$

Forming and Evaluating Composite Functions

A composite function $f \circ g$ is formed by applying one function to the result of another: $f(g(x))$.

• To evaluate $f(g(x))$, substitute $g(x)$ into $f$.

• Example: If $f(x) = 4x + 1$ and $g(x) = \frac{x-2}{x}$, then $f(g(x)) = 4\left(\frac{x-2}{x}\right) + 1 = \frac{4x-8}{x} + 1 = \frac{4x-8 + x}{x} = \frac{5x-8}{x}$.

Applications of Functions

Function Interpretation and Real-World Meaning

Functions can be used to model real-world relationships, such as converting between units.

• Example: $F(C) = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit.

• Evaluating $F(100)$ gives the Fahrenheit equivalent of 100°C: $F(100) = \frac{9}{5} \times 100 + 32 = 180 + 32 = 212$.

• The inverse function $C(F) = \frac{5}{9}(F - 32)$ converts Fahrenheit to Celsius.

One-to-One Functions

Definition of a One-to-One Function

A function is one-to-one if each output value corresponds to exactly one input value.

• Formal definition: $f(u) = f(v)$ implies $u = v$ for all $u, v$ in the domain of $f$.

• Alternate definition: No two different inputs produce the same output.

Examples of One-to-One and Not One-to-One Functions

• One-to-one example: $f(x) = x$ (each x maps to a unique y).

• Not one-to-one example: $f(x) = x^2$ (both $x = 2$ and $x = -2$ map to $y = 4$).

Horizontal Line Test

The Horizontal Line Test is a graphical method to determine if a function is one-to-one.

• If every horizontal line intersects the graph at most once, the function is one-to-one.

• Example: The graph of $f(x) = x^3$ passes the test; $f(x) = x^2$ does not.

Inverse Functions

Definition of an Inverse Function

An inverse function $f^{-1}(x)$ reverses the effect of $f(x)$, mapping outputs back to their original inputs.

• Formal definition: $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$ and $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$.

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

Composition Cancellation Equations

• $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$

• $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$

Verifying Inverse Functions

• To verify if $g(x)$ is the inverse of $f(x)$, check both composition cancellation equations.

• Example: If $f(x) = 3x - 5$ and $g(x) = \frac{x+5}{3}$, then $f(g(x)) = 3\left(\frac{x+5}{3}\right) - 5 = x + 5 - 5 = x$ and $g(f(x)) = \frac{3x-5+5}{3} = x$.

Graphing Inverse Functions

Sketching the Graph of an Inverse Function

The graph of an inverse function is the reflection of the original function's graph across the line $y = x$.

• Points $(a, b)$ on the graph of $f(x)$ correspond to points $(b, a)$ on the graph of $f^{-1}(x)$.

• If the two functions have only points in common, those points must lie on the line $y = x$.

Summary Table: Properties of Functions and Their Inverses

Property	Function $f(x)$	Inverse $f^{-1}(x)$
Domain	Set of valid inputs for $f$	Range of $f$
Range	Set of possible outputs of $f$	Domain of $f$
One-to-One	Required for inverse to be a function	Always one-to-one
Graph	Original graph	Reflection across $y = x$

Additional info: These notes expand on handwritten and brief points, providing formal definitions, examples, and academic context for College Algebra students studying functions, one-to-one functions, and inverse functions.