College Algebra: Functions, Equations, Graphs, and Applications Study Guide

Functions and Their Graphs

Definition and Evaluation of Functions

A function is a relation that assigns exactly one output value for each input value. Functions are often written as f(x), where x is the input variable.

• Evaluating a function: Substitute the given value into the function and simplify.

• Example: If $g(x) = 4x - 2$, then $g(a + 1) = 4(a + 1) - 2 = 4a + 4 - 2 = 4a + 2$.

Graphing Functions

To graph a function, plot points for various input values and connect them smoothly if the function is continuous.

• Example: For $f(x) = \frac{1}{2}x + 4$, plot points for several x-values and draw the line.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values).

• Example: For $f(x) = \frac{x}{x-4}$, the domain is all real numbers except $x = 4$ (since division by zero is undefined).

Interval Notation

Interval notation is used to describe sets of numbers, such as domains and ranges.

• Example: The interval $(-\infty, 4) \cup (4, \infty)$ represents all real numbers except 4.

Linear Equations and Applications

Slope and Intercepts

The slope of a line measures its steepness and is calculated as the change in y divided by the change in x between two points.

• Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• y-intercept: The point where the line crosses the y-axis ($x = 0$).

• x-intercept: The point where the line crosses the x-axis ($y = 0$).

Equation of a Line

The slope-intercept form of a line is $y = mx + b$, where m is the slope and b is the y-intercept.

• Example: A line passing through (5, -3) and (7, -2): $m = \frac{-2 - (-3)}{7 - 5} = \frac{1}{2}$ Use point-slope form: $y - (-3) = \frac{1}{2}(x - 5)$, then solve for y.

Linear Models and Applications

Linear functions can model real-world situations, such as cost, distance, or value over time.

• Example: The cost of a cab ride: $y = 3.14 + 1.43x$, where x is the number of miles.

• Example: The value of a stock: $f(x) = -0.61x + 1289.9$, where x is the year with 1990 as x = 2000.

Relations and Functions

Determining if a Relation is a Function

A relation is a function if each input value corresponds to exactly one output value.

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

• Example: The set of ordered pairs {(x, y)} is a function if no x-value repeats with different y-values.

Solving Equations and Inequalities

Solving Linear Equations

To solve a linear equation, isolate the variable using algebraic operations.

• Example: $\frac{1}{4}(4x + 20) = \frac{1}{2}(x - 12)$ Multiply both sides by 4 to clear denominators, then solve for x.

Classifying Equations

• Identity: An equation true for all values of the variable.

• Contradiction: An equation with no solution.

• Conditional: An equation true for some values of the variable.

Solving Inequalities

To solve inequalities, use similar steps as equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

• Example: $-3(x - 5) \leq 24$ Solve for x and express the solution in interval notation.

Factoring Expressions

Factoring Quadratic Expressions

Factoring is the process of writing an expression as a product of its factors.

• Example: $x^2 + 8x + 16 = (x + 4)^2$

• Example: $42x^2 - 17x - 6x^2 = 36x^2 - 17x$ (combine like terms before factoring)

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions for different intervals of the domain.

• Example: $f(x) = \begin{cases} x - 5 & x \geq 0 \\ -3 & x < 0 \end{cases}$

Applications and Word Problems

Setting Up Functions from Context

Translate real-world scenarios into algebraic functions.

• Example: If Bob's car gets 11 miles per gallon, the number of miles y he can travel with x gallons is $y = 11x$.

Mixture Problems

Mixture problems involve combining solutions of different concentrations to achieve a desired concentration.

• Example: Mixing 3 liters of 4% solution with x liters of 10% solution to get a 6% solution: $0.04 \times 3 + 0.10 \times x = 0.06 \times (3 + x)$

Geometry Applications

Use algebraic equations to solve geometric problems, such as perimeter and area.

• Example: The length of a rectangle with perimeter 140 m, if the length is 6 m more than the width: Let width = w, length = w + 6, then $2w + 2(w + 6) = 140$

Tables: Comparison and Classification

Cab Ride Cost Table

The table compares the cost of a cab ride for different distances.

x (miles)	y (cost)
1	3.14
2	4.43
3	5.72
4	7.01
5	8.30
6	9.59

Main purpose: To show the relationship between distance traveled and cost, illustrating a linear function.

Function Table

The table lists x-values and corresponding y-values to determine if the relation is a function.

x	y
-3	1.3
-2	2.4
-1	2.7
0	3.0
1	3.4
2	3.9
3	4.4
4	4.9

Main purpose: To check if each x-value corresponds to only one y-value, confirming the definition of a function.

Additional info:

• Some questions involve graph interpretation, such as identifying intervals where a function is increasing or decreasing.

• Compound inequalities and their graphical solutions are included.

• Factoring problems cover both quadratic and difference of squares forms.