College Algebra Study Notes: Functions, Graphs, and Extrema

Functions and Their Graphs

Evaluating Functions from Graphs

Understanding how to interpret and extract information from the graph of a function is a fundamental skill in College Algebra. This includes evaluating function values, determining domains and ranges, and identifying intercepts.

• Function Value: The value of a function at a specific input, $f(a)$, is found by locating $x = a$ on the graph and reading the corresponding $y$-value.

• Domain: The set of all possible $x$-values for which the function is defined.

• Range: The set of all possible $y$-values the function can take.

• Intercepts: Points where the graph crosses the axes. x-intercepts occur where $f(x) = 0$; y-intercepts occur where $x = 0$.

Example: Given a graph, you may be asked to find $f(0)$, $f(\frac{\pi}{6})$, and $f(\frac{7\pi}{2})$, or to determine the domain and range by observing the extent of the graph along the $x$ and $y$ axes.

Solving Equations and Inequalities from Graphs

Graphs can be used to solve equations such as $f(x) = a$ or inequalities like $f(x) > 0$ by identifying the $x$-values where the graph meets certain $y$-values.

• To solve $f(x) = a$: Find all $x$ where the graph crosses the horizontal line $y = a$.

• To solve $f(x) > 0$: Identify intervals where the graph is above the $x$-axis.

Example: For what values of $x$ does $f(x) = -2$? For what values of $x$ is $f(x) > 0$?

Analyzing Functions Algebraically

Domain of Rational Functions

The domain of a rational function $f(x) = \frac{p(x)}{q(x)}$ is all real numbers except where the denominator $q(x) = 0$.

• Example: For $f(x) = \frac{x+3}{x+4}$, the domain is $\{x \mid x \neq -4\}$.

Finding Points on the Graph

To determine if a point $(a, b)$ is on the graph of $f(x)$, check if $f(a) = b$.

• Example: Is $(\frac{1}{4}, 1)$ on the graph of $f(x) = \frac{x+3}{x+4}$? Substitute $x = \frac{1}{4}$ and check if $f(\frac{1}{4}) = 1$.

Finding Intercepts

• x-intercepts: Set $f(x) = 0$ and solve for $x$.

• y-intercept: Evaluate $f(0)$.

Applications: Cost Functions

Modeling with Functions

Functions can model real-world scenarios, such as the cost per passenger for a plane crossing the Atlantic Ocean. The cost function might be given by:

• $C(x) = 150 + \frac{38,000}{x}$, where $x$ is the ground speed in miles per hour.

Key Points:

• To find the cost at a specific speed, substitute $x$ into the function.

• The domain is restricted to $x > 0$ (since speed must be positive).

• Graphing and tabulating values helps identify minimum cost.

x (mph)	C(x)
450	234.44
500	226.00
550	219.09
600	213.33

Example: The minimum cost per passenger occurs at approximately 550 miles per hour.

Even and Odd Functions

Definitions

• 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.

Theorem:

• A function is even if and only if its graph is symmetric with respect to the $y$-axis.

• A function is odd if and only if its graph is symmetric with respect to the origin.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd; $f(x) = x^3 + x$ is neither.

Identifying Even and Odd Functions from Graphs and Algebraically

• Check symmetry visually or by substituting $-x$ into the function.

• If $f(-x) = f(x)$, the function is even.

• If $f(-x) = -f(x)$, the function is odd.

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

Increasing, Decreasing, and Constant Functions

Definitions

• Increasing: $f$ is increasing on an interval if for any $x_1 < x_2$, $f(x_1) < f(x_2)$.

• Decreasing: $f$ is decreasing on an interval if for any $x_1 < x_2$, $f(x_1) > f(x_2)$.

• Constant: $f$ is constant on an interval if $f(x_1) = f(x_2)$ for all $x_1, x_2$ in the interval.

Example: From a graph, identify intervals where the function is increasing, decreasing, or constant by observing the slope as you move from left to right.

Local and Absolute Extrema

Definitions

• Local Maximum: $f(c)$ is a local maximum if $f(c) \geq f(x)$ for all $x$ near $c$.

• Local Minimum: $f(c)$ is a local minimum if $f(c) \leq f(x)$ for all $x$ near $c$.

• Absolute Maximum: $f(c)$ is the largest value of $f(x)$ on its domain.

• Absolute Minimum: $f(c)$ is the smallest value of $f(x)$ on its domain.

Finding Extrema from Graphs:

• Local maxima and minima are found at peaks and valleys of the graph.

• Absolute extrema are the highest and lowest points on the entire graph (within the domain).

Example: For a function $f(x)$ on $[a, b]$, the absolute maximum is the largest $f(x)$ value on $[a, b]$, and the absolute minimum is the smallest $f(x)$ value on $[a, b]$.

Using Graphing Utilities

• Graphing calculators or software can approximate local maxima and minima.

• Use the graph to estimate where the function is increasing or decreasing.

Summary Table: Even vs. Odd Functions

Property	Even Function	Odd Function
Algebraic Test	$f(-x) = f(x)$	$f(-x) = -f(x)$
Graph Symmetry	y-axis	Origin
Example	$x^2$	$x^3$

Key Formulas

• Domain of Rational Function: $\{x \mid q(x) \neq 0\}$

• Cost Function Example: $C(x) = 150 + \frac{38,000}{x}$

• Even Function: $f(-x) = f(x)$

• Odd Function: $f(-x) = -f(x)$

Additional info: These notes expand on the provided materials by clarifying definitions, providing structured examples, and summarizing key properties and formulas relevant to College Algebra students.