Functions and Their Graphs: Key Concepts in Precalculus

Functions and Their Graphs

Objectives

This section introduces foundational concepts for analyzing functions and their graphs, which are essential in precalculus. Students will learn to:

• Identify intervals where a function increases, decreases, or remains constant.

• Locate relative maxima and minima using graphs.

• Test equations and graphs for symmetry.

• Recognize and classify even and odd functions.

• Understand and evaluate piecewise functions.

• Simplify a function's difference quotient.

Increasing, Decreasing, and Constant Functions

Definitions and Graphical Interpretation

Understanding how a function behaves over different intervals is crucial for graph analysis and calculus readiness.

• Increasing Function: A function is increasing on an open interval $I$ if, for any $x_1 < x_2$ in $I$, $f(x_1) < f(x_2)$.

• Decreasing Function: A function is decreasing on an open interval $I$ if, for any $x_1 < x_2$ in $I$, $f(x_1) > f(x_2)$.

• Constant Function: A function is constant on an open interval $I$ if, for any $x_1, x_2$ in $I$, $f(x_1) = f(x_2)$.

Graphical Features:

• Increasing: Graph rises from left to right.

• Decreasing: Graph falls from left to right.

• Constant: Graph is a horizontal line.

Example: Identifying Intervals

Given a function's graph, determine where it is increasing, decreasing, or constant.

• Example: For a sample cubic function, the graph increases on $(-\infty, -1)$ and $(1, \infty)$, and decreases on $(-1, 1)$.

Relative Maximum and Minimum

Definitions

Relative extrema are points where a function reaches a local highest or lowest value within a neighborhood.

• Relative Maximum: $f(a)$ is a relative maximum if there exists an open interval containing $a$ such that $f(a) \geq f(x)$ for all $x$ in that interval.

• Relative Minimum: $f(b)$ is a relative minimum if there exists an open interval containing $b$ such that $f(b) \leq f(x)$ for all $x$ in that interval.

• The term local is often used interchangeably with relative.

Example: Locating Relative Extrema

• On a graph, relative maxima and minima are typically found at peaks and valleys.

• Example: For a given function, $f$ has a relative maximum at $x = -1$ and a relative minimum at $x = 1$.

Symmetry of Graphs

Types of Symmetry and Tests

Symmetry helps classify functions and predict graph behavior.

Type of Symmetry	Definition	Test for Symmetry
y-axis	Graph is symmetric with respect to the y-axis if for every point $(x, y)$, the point $(-x, y)$ is also on the graph.	Substitute $-x$ for $x$ in the equation. If the equation is unchanged, the graph is y-axis symmetric.
x-axis	Graph is symmetric with respect to the x-axis if for every point $(x, y)$, the point $(x, -y)$ is also on the graph.	Substitute $-y$ for $y$ in the equation. If the equation is unchanged, the graph is x-axis symmetric.
Origin	Graph is symmetric with respect to the origin if for every point $(x, y)$, the point $(-x, -y)$ is also on the graph.	Substitute $-x$ for $x$ and $-y$ for $y$ in the equation. If the equation is unchanged, the graph is origin symmetric.

Example: Testing for Symmetry

• Given $y = x^3 + x$, substituting $-x$ for $x$ yields $y = (-x)^3 + (-x) = -x^3 - x$, which is not the same as the original equation, so the graph is not y-axis symmetric.

• Substituting $-y$ for $y$ does not yield the same equation, so not x-axis symmetric.

• Substituting both $-x$ and $-y$ does not yield the same equation, so not origin symmetric.

Even and Odd Functions

Definitions and Properties

Classifying functions as even or odd helps in understanding their symmetry and behavior.

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

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

Identifying Even or Odd Functions

• For an even function, replacing $x$ with $-x$ does not change the equation.

• For an odd function, replacing $x$ with $-x$ changes the sign of every term.

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

Example: $h(x) = x^4 + 1$

• $h(-x) = (-x)^4 + 1 = x^4 + 1 = h(x)$, so $h$ is even.

• For $h(x) = x^3 + 1$, $h(-x) = (-x)^3 + 1 = -x^3 + 1 \neq h(x)$ and $h(-x) \neq -h(x)$, so $h$ is neither even nor odd.

Piecewise Functions

Definition and Evaluation

A piecewise function is defined by different expressions over specified intervals of its domain.

• Each 'piece' applies to a certain interval.

• To evaluate, determine which interval the input belongs to and use the corresponding expression.

Example: If $C(x) = \begin{cases} 40 & x < 10 \\ 61 & x \geq 10 \end{cases}$, then $C(5) = 40$, $C(50) = 61$.

Graphing Piecewise Functions

• Graph each piece over its specified interval.

• Use open or closed dots to indicate whether endpoints are included.

• Combine all pieces for the complete graph.

Difference Quotient

Definition and Simplification

The difference quotient is a fundamental concept for understanding rates of change and the basis of derivatives in calculus.

• The difference quotient for a function $f$ is:

$\frac{f(x+h) - f(x)}{h}$

• To simplify, substitute $f(x)$ and $f(x+h)$, expand, and reduce the expression.

Example: For $f(x) = x^2$,

$\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Additional info: These concepts form the basis for further study in calculus, including limits, derivatives, and integrals.