Functions and Their Graphs: Precalculus Study Notes

Chapter 2: Functions and Their Graphs

Section 2.1: Functions

Describing a Relation

A relation is a correspondence between two sets, often represented as ordered pairs. The set of all first elements is called the domain, and the set of all second elements is called the range. Relations can be expressed in various forms and are foundational to understanding functions.

• Ways to Express a Relation:

  • Verbally

  • Numerically (table of values or set of ordered pairs)

  • Graphically (plotting points or mapping)

  • Algebraically (using an equation)

• Example: The relation between months and average temperature in Los Angeles:

  • Domain (D): {Jan, Feb, Mar, Apr, May, Jun}

  • Range (R): {54, 56, 61, 70, 75, 79}

  • Set of ordered pairs: {(Jan,54), (Feb,56), (Mar,61), (Apr,70), (May,75), (Jun,79)}

• Mapping: Shows how each element in the domain is paired with an element in the range.

Determining Whether a Relation Represents a Function

A function is a special type of relation where each element in the domain is paired with exactly one element in the range. The domain of a function is the set of all possible input values, and the range is the set of all possible output values.

• Definition: A function f from X to Y is a relation that associates each element of X (domain) with exactly one element of Y (range).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Examples:

  • y = x (line): Yes, it is a function.

  • y = x2 (parabola): Yes, it is a function.

  • x = y2 (sideways parabola): No, not a function.

  • x2 + y2 = 9 (circle): No, not a function.

Function Notation and Evaluating Functions

Function notation is a way to name functions and indicate the input variable. The independent variable is usually x, and the dependent variable is f(x).

• Function Notation: $f(x)$

• Evaluating Functions: Substitute the input value into the function.

  • Example: If $f(x) = 2x^2 - 5x$, then $f(2) = 2(2)^2 - 5(2) = 8 - 10 = -2$

  • For $f(-1)$: $f(-1) = 2(-1)^2 - 5(-1) = 2 - (-5) = 7$

• Calculator Use: Functions can be evaluated using a calculator for complex expressions.

Implicit and Explicit Forms of a Function

Functions can be expressed in explicit or implicit form. Explicit form solves for the dependent variable, while implicit form does not.

Implicit Form	Explicit Form
2x + y = 4	y = 4 - 2x
x2 = y	y = x2
y - 7 = 0	y = 7

The Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function and is foundational in calculus for defining derivatives.

• Definition: The difference quotient of a function $f$ is given by: $\frac{f(x+h) - f(x)}{h}$, where $h \neq 0$

• Example: For $f(x) = x^2 - 2$:

  • $f(x+h) = (x+h)^2 - 2 = x^2 + 2xh + h^2 - 2$

  • Difference quotient: $\frac{(x^2 + 2xh + h^2 - 2) - (x^2 - 2)}{h} = \frac{2xh + h^2}{h} = 2x + h$

Finding the Domain of a Function Defined by an Equation

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 even roots (radicands must be non-negative).

• Steps to Find the Domain:

  1. If the function has a denominator, exclude values that make the denominator zero.

  2. If the function has an even root, exclude values that make the radicand negative.

• Examples:

  • $f(x) = \frac{5x}{x^2 - 2}$: Exclude $x$ such that $x^2 - 2 = 0 \Rightarrow x = \sqrt{2}, -\sqrt{2}$

  • $f(x) = \sqrt{x - 2}$: Domain is $x \geq 2$

  • $f(x) = \frac{x+5}{x-5}$: Exclude $x = 5$

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function is the intersection of the domains of the original functions, with additional restrictions for division.

• Sum Function: $(f+g)(x) = f(x) + g(x)$

• Difference Function: $(f-g)(x) = f(x) - g(x)$

• Product Function: $(fg)(x) = f(x) \cdot g(x)$

• Quotient Function: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Operation	Definition	Domain
Sum	$(f+g)(x) = f(x) + g(x)$	Intersection of domains of $f$ and $g$
Difference	$(f-g)(x) = f(x) - g(x)$	Intersection of domains of $f$ and $g$
Product	$(fg)(x) = f(x) \cdot g(x)$	Intersection of domains of $f$ and $g$
Quotient	$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$	Intersection of domains of $f$ and $g$, $g(x) \neq 0$

Example: Operations on Functions

• Let $f(x) = \frac{1}{x+1}$ and $g(x) = \frac{x}{x^2 - 3}$

• Find $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\left(\frac{f}{g}\right)(x)$

• Determine the domain for each operation by considering restrictions from both $f$ and $g$.

Summary: Understanding relations, functions, their domains, and operations is essential for further study in precalculus and calculus. Mastery of these concepts allows students to analyze and manipulate mathematical models effectively.