Precalculus Study Notes: Functions, Domains, Ranges, and Composition

Functions: Domain, Range, and Evaluation

Definition of a Function

A function is a relation that assigns each element in the domain to exactly one element in the range. Functions are often denoted as f(x), where x is the input variable.

• Domain: The set of all possible input values (x) for which the function is defined.

• Range: The set of all possible output values (f(x)) that the function can produce.

• Function Evaluation: To evaluate a function at a specific value, substitute the value into the function's formula.

Example: If $f(x) = x^2$, then $f(3) = 3^2 = 9$.

Piecewise Functions

A piecewise function is defined by different expressions depending on the value of the input variable.

• Each 'piece' applies to a specific interval of the domain.

• To evaluate, determine which interval the input falls into and use the corresponding expression.

Example: $k(x) = \begin{cases} 2x-1, & x < 1 \\ 3, & x = 1 \\ x+2, & x > 1 \end{cases}$

Finding Domain and Range

• For polynomial functions, the domain is usually all real numbers.

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

• For square root functions, the radicand must be non-negative.

• For piecewise functions, consider the union of intervals defined by each piece.

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

Function Composition

Definition and Notation

The composition of functions combines two functions such that the output of one function becomes the input of another. The notation is $(f \circ g)(x) = f(g(x))$.

• Order matters: $f(g(x))$ is generally not equal to $g(f(x))$.

• Domain of Composition: The domain of $f(g(x))$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Even and Odd Functions

Definitions

• Even Function: $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric about the y-axis.

• Odd Function: $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric about the origin.

• Neither: If neither condition is met, the function is neither even nor odd.

Example: $f(x) = x^2$ is even because $(-x)^2 = x^2$. $f(x) = x^3$ is odd because $(-x)^3 = -x^3$.

Restrictions on Domain

Common Restrictions

• Division by zero is undefined.

• Even roots (such as square roots) require non-negative radicands.

• Logarithms require positive arguments.

Example: For $f(x) = \sqrt{x-4}$, the domain is $x \geq 4$.

Tables: Function Values and Composition

Purpose

Tables are used to organize function values and to facilitate evaluation and composition.

x	f(x)	g(x)
-3	1	2
-1	2	4
0	4	1
2	5	3
4	3	4

Example: To find $f(g(0))$, first find $g(0)$ from the table (which is 1), then find $f(1)$ using the function or table.

Piecewise Function Table Example

x	h(x)
x < 2	$\frac{1}{x}$
x = 2	1
x > 3	4

Example: For $x = 1$, $h(1) = \frac{1}{1} = 1$.

Summary of Key Concepts

• Understand how to determine the domain and range of various types of functions.

• Be able to evaluate functions and their compositions using formulas, tables, and graphs.

• Recognize and classify functions as even, odd, or neither based on their symmetry properties.

• Apply restrictions to domains based on the type of function (rational, root, logarithmic, etc.).

• Use tables to organize and compute function values and compositions.

Additional info: Some context and examples were inferred to provide a complete and self-contained study guide based on the worksheet structure and standard Precalculus curriculum.