College Algebra: Functions, Graphs, and Equations Study Guide

Functions and Relations

Definition and Identification

Understanding the concept of a function is fundamental in College Algebra. A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Relation: A set of ordered pairs.

• Function: A relation where no two ordered pairs have the same first element and different second elements.

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

• Example: The set {(1, -4), (2, -11), (6, -5), (8, 3), (10, 30)} is a function because each x-value is unique.

Evaluating Functions

Substitution and Calculation

To evaluate a function, substitute the given value into the function's formula.

• Example: For $f(x) = x^2 - 2x - 7$, find $f(3)$:

  • Substitute $x = 3$:

  • $f(3) = 3^2 - 2(3) - 7 = 9 - 6 - 7 = -4$

• Example: For $f(x) = 4x^2 + 3x - 7$, find $f(2)$:

  • $f(2) = 4(2)^2 + 3(2) - 7 = 16 + 6 - 7 = 15$

Transformations of Functions

Shifting, Reflecting, and Stretching

Transformations change the position or shape of a graph. Common transformations include shifts, reflections, and stretches.

• Horizontal Shift: $f(x - h)$ shifts the graph $h$ units to the right; $f(x + h)$ shifts $h$ units to the left.

• Vertical Shift: $f(x) + k$ shifts the graph $k$ units up; $f(x) - k$ shifts $k$ units down.

• Reflection: $-f(x)$ reflects the graph across the x-axis; $f(-x)$ reflects across the y-axis.

• Example: To obtain $f(x) = - rac{1}{2}x^2 + 4$ from $y = x^2$, reflect across the x-axis, vertically stretch by $rac{1}{2}$, and shift up by 4 units.

Graphs of Functions

Function Graphs and Their Properties

Graphs visually represent functions and their behavior. Key properties include symmetry, intercepts, and whether the graph passes the vertical line test.

• Symmetry: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• Intercepts: Points where the graph crosses the axes.

• Example: The graph of $y = x^2$ is symmetric about the y-axis (even function).

Inverse Functions

Definition and Verification

Two functions $f$ and $g$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in their domains.

• Example: $f(x) = rac{3x}{x^2 - 2}$ and $g(x) = rac{2x + 3}{x}$; check if $f(g(x)) = x$ and $g(f(x)) = x$.

Operations on Functions

Sum, Difference, Product, and Quotient

Functions can be combined using arithmetic operations.

• Sum: $(f + g)(x) = f(x) + g(x)$

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

• Product: $(f imes g)(x) = f(x) imes g(x)$

• Quotient: $(f / g)(x) = rac{f(x)}{g(x)}$, $g(x) eq 0$

• Example: If $f(x) = x + 5$, $g(x) = x - 4$, then $(f + g)(x) = (x + 5) + (x - 4) = 2x + 1$

Piecewise Functions

Definition and Evaluation

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

• Example: $f(x) = \begin{cases} 6 & \text{if } x \leq -1 \\ 4x & \text{if } 0 < x < 3 \\ x & \text{if } x \geq 3 \end{cases}$

• To evaluate, choose the appropriate case based on the value of $x$.

Linear Equations and Graphs

Slope, Point-Slope, and Slope-Intercept Forms

Linear equations can be written in several forms, each useful for different purposes.

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

• Example: Find the equation of the line through $(3, 5)$ with slope $-3$:

  • Point-slope: $y - 5 = -3(x - 3)$

  • Slope-intercept: $y = -3x + 14$

Parallel and Perpendicular Lines

Equations Under Given Conditions

Lines are parallel if they have the same slope, and perpendicular if their slopes are negative reciprocals.

• Parallel: If a line has equation $y = mx + b$, a parallel line through $(x_1, y_1)$ is $y - y_1 = m(x - x_1)$.

• Perpendicular: The slope of a perpendicular line is $-1/m$.

• Example: Find the equation of the line passing through $(2, 4)$ and perpendicular to $4x + y = 3$:

  • First, rewrite $4x + y = 3$ as $y = -4x + 3$ (slope $-4$).

  • Perpendicular slope is $1/4$.

  • Equation: $y - 4 = \frac{1}{4}(x - 2)$

Distance Between Two Points

Distance Formula

The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Between $(7, 7)$ and $(-6, -7)$:

  • $d = \sqrt{(7 - (-6))^2 + (7 - (-7))^2} = \sqrt{13^2 + 14^2} = \sqrt{169 + 196} = \sqrt{365}$

Equations and Graphs of Circles

Standard Form and Graphing

The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• Example: Center $(-8, 6)$, radius $3$:

  • Equation: $(x + 8)^2 + (y - 6)^2 = 9$

Even and Odd Functions

Definitions and Tests

Functions can be classified as even, odd, or neither based on their symmetry.

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis).

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

One-to-One Functions

Definition and Determination

A function is one-to-one if each output is produced by exactly one input; that is, $f(x_1) = f(x_2)$ implies $x_1 = x_2$.

• Horizontal Line Test: A graph is one-to-one if no horizontal line intersects it more than once.

• Example: $f(x) = x^3$ is one-to-one; $f(x) = x^2$ is not.

True/False Statements about Circles

Center and Equation

To determine the center of a circle from its equation, compare to the standard form $(x - h)^2 + (y - k)^2 = r^2$.

• Example: The equation $(x + 8)^2 + (y - 6)^2 = 9$ has center $(-8, 6)$, not $(8, 8)$.

Summary Table: Function Properties

Property	Test	Example
Function	Vertical Line Test	$y = x^2$
One-to-One	Horizontal Line Test	$y = x^3$
Even	$f(-x) = f(x)$	$y = x^2$
Odd	$f(-x) = -f(x)$	$y = x^3$

Additional info: Some explanations and examples have been expanded for clarity and completeness beyond the original questions.