College Algebra: Study Guide for Test 1 (Chapters 1 & 2)

Chapter 1: Introduction to Functions and Graphs

Evaluating Functions

To evaluate a function, substitute the specified value or expression for the variable in the function's formula.

• Definition: A function is a rule that assigns to each input exactly one output.

• Example: If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$.

Intercepts of a Function

Intercepts are points where the graph of a function crosses the axes.

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

• y-intercept: Evaluate $f(0)$.

• Example: For $f(x) = x^2 - 4$, x-intercepts: $x^2 - 4 = 0 \implies x = \pm2$; y-intercept: $f(0) = -4$.

Determining Functions and Relations

Not all relations are functions. A function assigns exactly one output to each input.

• Vertical Line Test: If any vertical line crosses the graph more than once, it is not a function.

• Example: The graph of $y = x^2$ passes the vertical line test; $x = y^2$ does not.

Domain and Range

The domain is the set of all possible input values; the range is the set of all possible output values.

• Symbolic: Exclude values that make denominators zero or radicands of even roots negative.

• Numeric/Table: Domain: all listed $x$-values; Range: all listed $f(x)$-values.

• Graphical: Domain: projection onto $x$-axis; Range: projection onto $y$-axis.

• Example: $f(x) = \frac{1}{x-2}$ has domain $x \neq 2$.

Applications of Functions

Functions are used to model real-world relationships, such as cost, distance, or population over time.

• Example: If $C(x) = 5x + 20$ models the cost $C$ for $x$ items, then $C(10) = 5(10) + 20 = 70$.

Linear vs. Nonlinear Functions

Linear functions graph as straight lines; nonlinear functions do not.

• Linear: $f(x) = mx + b$

• Nonlinear: $f(x) = x^2$, $f(x) = \sqrt{x}$, etc.

• Example: $f(x) = 3x - 2$ is linear; $f(x) = x^2 + 1$ is nonlinear.

Intervals of Increase, Decrease, and Constancy

Describe where a function's graph rises, falls, or remains flat.

• Increasing: $f(x_2) > f(x_1)$ for $x_2 > x_1$ in the interval.

• Decreasing: $f(x_2) < f(x_1)$ for $x_2 > x_1$ in the interval.

• Constant: $f(x_2) = f(x_1)$ for all $x_2, x_1$ in the interval.

• Example: $f(x) = x^2$ is decreasing on $(-\infty, 0)$ and increasing on $(0, \infty)$.

Chapter 2: Linear Functions and Equations

Equations of Lines

Lines can be written in several forms, each useful in different contexts.

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Vertical Line: $x = a$

• Horizontal Line: $y = b$

• Example: Line through $(2,3)$ with slope $-1$: $y - 3 = -1(x - 2)$

Graphing Linear Equations

To graph a line, plot the y-intercept and use the slope to find another point.

• Example: For $y = 2x + 1$, y-intercept is $(0,1)$; slope is $2$ (rise 2, run 1).

Linear Modeling

Linear models represent real-world relationships with a straight-line equation.

• Steps: Identify variables, determine slope and intercept, write equation.

• Example: If a taxi charges $3$plus $2$ per mile: $C = 2m + 3$.

Solving Linear Equations

Linear equations can have one solution (conditional), all real numbers (identity), or no solution (contradiction).

• Conditional: $2x = 6 \implies x = 3$

• Identity: $x + 2 = x + 2$ (all $x$)

• Contradiction: $x + 2 = x + 3$ (no solution)

Word Problems: Linear Equations and Inequalities

Translate real-world scenarios into equations or inequalities, then solve.

• Example: "A number plus 5 is 12." $x + 5 = 12 \implies x = 7$

Intersection of Graphs Method

To solve equations or inequalities, graph both sides and find intersection points.

• Example: Solve $2x + 1 = x + 4$ by graphing $y_1 = 2x + 1$ and $y_2 = x + 4$; intersection at $x = 3$.

Solving Linear Inequalities

Find all $x$ that make the inequality true; express solutions in interval or set-builder notation.

• Example: $2x - 3 < 5 \implies 2x < 8 \implies x < 4$

• Interval Notation: $(-\infty, 4)$

• Set-Builder Notation: $\{x \mid x < 4\}$

Piecewise-Defined Functions

Functions defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} x^2, & x < 0 \\ x + 1, & x \geq 0 \end{cases}$

• Domain: All real numbers, unless otherwise restricted.

Direct Variation

In direct variation, one variable is a constant multiple of another.

• Form: $y = kx$, where $k$ is the constant of variation.

• Example: If $y$ varies directly as $x$ and $y = 10$ when $x = 2$, then $k = 5$ and $y = 5x$.