College Algebra: Study Guide on Functions, Transformations, and Operations

Functions and Their Properties

Definition and Basic Concepts

A function is a relation that assigns exactly one output value for each input value from a given domain. Functions are fundamental objects in algebra and are used to model relationships between quantities.

• Domain: The set of all possible input values (usually x-values) for which the function is defined.

• Range: The set of all possible output values (usually y-values) that the function can produce.

• Function Notation: If f is a function, then f(x) denotes the output when the input is x.

Graphing Functions

Graphing a function involves plotting points (x, f(x)) on a coordinate plane. The graph visually represents the relationship between the input and output values.

• Parent Functions: Basic functions such as linear, quadratic, cubic, absolute value, square root, and reciprocal functions serve as building blocks for more complex functions.

• Transformations: Changes to the graph of a function, including translations (shifts), reflections, stretches, and compressions.

Example: Graphing a Linear Function

• Given the equation $y = 2x + 1$, plot several points by choosing x-values and calculating the corresponding y-values.

• Draw a straight line through the plotted points.

Linear Equations and Their Graphs

Standard and Slope-Intercept Forms

• Standard Form: $Ax + By = C$

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

Finding Equations of Lines

• To find the equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, first calculate the slope:

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

• Then use the point-slope form:

$y - y_1 = m(x - x_1)$

• Convert to slope-intercept or standard form as needed.

Parallel and Perpendicular Lines

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals: if one line has slope m, the other has slope $-1/m$.

Transformations of Functions

Types of Transformations

• Translations: Shifting the graph horizontally or vertically.

• Reflections: Flipping the graph over a line, such as the x-axis or y-axis.

• Stretches and Compressions: Changing the steepness or width of the graph.

Example: Transforming a Function

• Given $f(x) = x^2$, the graph of $f(x - 2) + 3$ is the graph of $x^2$ shifted right by 2 units and up by 3 units.

Operations with Functions

Function Operations

• Addition: $(f + g)(x) = f(x) + g(x)$

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

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$

Example: Function Addition

• If $f(x) = x^2$ and $g(x) = 2x$, then $(f + g)(x) = x^2 + 2x$.

Composition and Inverses of Functions

Function Composition

The composition of two functions $f$ and $g$ is written as $(f \circ g)(x) = f(g(x))$.

• To evaluate, first apply $g$ to $x$, then apply $f$ to the result.

Example: Function Composition

• If $f(x) = 2x$ and $g(x) = x + 3$, then $(f \circ g)(x) = f(g(x)) = 2(x + 3) = 2x + 6$.

Inverse Functions

• A function $f$ has an inverse $f^{-1}$ if and only if it is one-to-one (each output is produced by exactly one input).

• The inverse function "undoes" the action of the original function: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Example: Finding an Inverse

• Given $f(x) = 3x + 2$, solve for $x$ in terms of $y$:

$y = 3x + 2$ $x = \frac{y - 2}{3}$ So, $f^{-1}(x) = \frac{x - 2}{3}$

Difference Quotient

Definition and Simplification

The difference quotient is a formula that measures the average rate of change of a function over an interval. It is foundational for calculus.

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

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

Example: Difference Quotient for $f(x) = x^2$

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

Summary Table: Function Operations and Properties

Operation	Definition	Example
Addition	$(f + g)(x) = f(x) + g(x)$	$(x^2) + (2x) = x^2 + 2x$
Subtraction	$(f - g)(x) = f(x) - g(x)$	$(x^2) - (2x) = x^2 - 2x$
Multiplication	$(f \cdot g)(x) = f(x) \cdot g(x)$	$(x^2) \cdot (2x) = 2x^3$
Division	$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$	$\frac{x^2}{2x} = \frac{x}{2}$ (for $x \neq 0$)
Composition	$(f \circ g)(x) = f(g(x))$	$f(x) = 2x$, $g(x) = x + 3$, so $f(g(x)) = 2(x + 3) = 2x + 6$

Additional info:

• Some content and examples were inferred to provide a complete and self-contained study guide, as the original file contained only brief prompts and questions.