Functions: Definitions, Domains, and Difference Quotients in Precalculus

Functions and Their Properties

Definitions

Understanding the concept of a function is fundamental in precalculus. Functions relate each input to exactly one output, and their domains and ranges describe the set of possible inputs and outputs, respectively.

• Function: A function assigns each input number to one output number.

• Domain: The set of all input numbers for which the function is defined.

• Range: The set of all possible output numbers produced by the function.

• Equality of Functions: Two functions f and g are equal if:

  1. Their domains are equal.

  2. For every input in the domain, $f(x) = g(x)$.

Determining Equality of Functions

Example 1: Which Functions Are Equal?

To determine if two functions are equal, compare both their domains and their output values for each input.

• Key Point: Functions may appear different but can be equal if their domains and outputs match for all inputs.

• Example:

  • $f(x) = \frac{x^2(x-1)}{x^2-1}$

  • $g(x) = x^2$

  • $h(x) = \begin{cases} x^2 & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \end{cases}$

  • $k(x) = \begin{cases} x^2 & \text{if } x \neq 1 \\ 3 & \text{if } x = 1 \end{cases}$

  Analysis: Compare the domains and outputs for $x = 1$ and $x \neq 1$ to determine equality.

Domain of Functions

Example 2: Finding the Domain

The domain of a function is the set of all input values for which the function is defined. Restrictions may arise from denominators (cannot be zero) or square roots (argument must be non-negative).

• Key Point: For rational functions, exclude values that make the denominator zero.

• Key Point: For functions involving square roots, the radicand must be greater than or equal to zero.

• Examples:

  • $f(x) = \frac{x}{x^2 - x - 2}$ Domain: All real $x$ except where $x^2 - x - 2 = 0$ (i.e., $x = 2$ or $x = -1$).

  • $g(x) = \sqrt{2x - 1}$ Domain: $2x - 1 \geq 0 \implies x \geq \frac{1}{2}$.

Domain and Function Values

Example 3: Evaluating Functions and Their Domains

To evaluate a function at a specific value, substitute the input into the function's formula. The domain is determined by the set of all permissible inputs.

• Key Point: Polynomial functions have domains of all real numbers unless otherwise restricted.

• Example: For $g(x) = 3x^2 + 5$, the domain is all real numbers.

• Application:

  • Find $g(1)$: Substitute $x = 1$ into $g(x)$.

  • Find $g(7)$: Substitute $x = 7$ into $g(x)$.

  • Find $g(x + h)$: Substitute $x + h$ into $g(x)$.

Difference Quotient

Example 4: Calculating the Difference Quotient

The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval. It is defined as:

• $\frac{f(x+h) - f(x)}{h}$

• Key Point: The difference quotient is used to find the slope of the secant line between two points on the graph of a function.

• Examples:

  • For $f(x) = x^2$: $\frac{(x+h)^2 - x^2}{h}$

  • For $f(x) = \frac{x}{3}$: $\frac{\frac{x+h}{3} - \frac{x}{3}}{h}$