Functions and Graphs: Precalculus Study Notes

Ch. 2 – Functions and Graphs

Introduction to Functions and Their Graphs

This section introduces the foundational concepts of relations and functions, their graphical representations, and how to determine if a relation is a function. Understanding these concepts is essential for analyzing mathematical models and real-world phenomena.

• Relation: A connection between two sets of values, often represented as ordered pairs (x, y).

• Function: A special type of relation where each input (x-value) has exactly one output (y-value).

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

Example: The set {(1, 2), (2, 3), (3, 4)} is a function, but {(1, 2), (1, 3)} is not.

Inputs and Outputs

Functions can be visualized as machines that assign each input exactly one output.

• Input: The independent variable, usually x.

• Output: The dependent variable, usually y or f(x).

Verifying if Equations are Functions

To determine if an equation is a function, check if each input yields only one output.

• Given a graph: Use the vertical line test.

• Given an equation: Solve for y in terms of x and check if each x gives only one y.

Example: $y = x^2$ is a function; $y^2 = x$ is not (since one x can yield two y values).

Domain and Range of a Graph

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

• Domain: All allowed x-values.

• Range: All allowed y-values.

• Interval Notation: Uses parentheses ( ) for exclusion and brackets [ ] for inclusion.

• Set Builder Notation: Describes the set using inequalities, e.g., $\{x | -2 \leq x < 5\}$.

Example: For $f(x) = \sqrt{x}$, domain is $[0, \infty)$.

Finding the Domain of an Equation

To find the domain algebraically, identify restrictions such as:

• Square Roots: The expression under the root must be non-negative.

• Denominators: The denominator cannot be zero.

Example: For $f(x) = \sqrt{x-2}$, domain is $[2, \infty)$.

Example: For $f(x) = \frac{1}{x-3}$, domain is $(-\infty, 3) \cup (3, \infty)$.

Graphs of Common Functions

Several basic functions frequently appear in Precalculus. Their domains and ranges are important to know.

Function	Equation	Domain	Range
Constant	$f(x) = c$	$(-\infty, \infty)$	$\{c\}$
Identity	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$
Square	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cube	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Cube Root	$f(x) = \sqrt[3]{x}$	$(-\infty, \infty)$	$(-\infty, \infty)$

Transformations of Functions

Transformations change the position and/or shape of a function's graph. The main types are reflections, shifts, and stretches/shrinks.

• Reflection: Flips the graph over the x-axis ($f(x) \to -f(x)$) or y-axis ($f(x) \to f(-x)$).

• Vertical Shift: $f(x) + k$ shifts up if $k > 0$, down if $k < 0$.

• Horizontal Shift: $f(x - h)$ shifts right if $h > 0$, left if $h < 0$.

• Vertical Stretch/Shrink: $a f(x)$ stretches if $|a| > 1$, shrinks if $0 < |a| < 1$.

• Horizontal Stretch/Shrink: $f(bx)$ shrinks if $|b| > 1$, stretches if $0 < |b| < 1$.

Example: $f(x) = x^2$ reflected over the x-axis is $-x^2$; shifted right 3 units is $(x-3)^2$.

Domain and Range of Transformed Functions

Transformations affect the domain and range of functions. Shifts move the domain/range, while stretches/shrinks scale them.

• For $g(x) = f(x-h) + k$, domain is shifted by $h$, range by $k$.

Example: If $f(x)$ has domain $[0, 4]$, then $g(x) = f(x-2) + 3$ has domain $[2, 6]$ and range shifted up by 3.

Function Operations

Functions can be added, subtracted, multiplied, or divided. The domain of the resulting function is the intersection of the domains of the original functions (and for division, where the denominator is not zero).

• Addition: $(f+g)(x) = f(x) + g(x)$

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

• Multiplication: $(fg)(x) = f(x)g(x)$

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

Example: If $f(x) = x^2$ and $g(x) = x+1$, $(f+g)(x) = x^2 + x + 1$.

Function Composition

Composition involves applying one function to the result of another: $(f \circ g)(x) = f(g(x))$.

• Evaluate inside out: first $g(x)$, then $f$ of that result.

• Domain: All x-values in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Decomposing Functions

Decomposition is expressing a function as a composition of two or more simpler functions.

• There are often multiple correct ways to decompose a function.

Example: $h(x) = \sqrt{2x-1}$ can be written as $h(x) = f(g(x))$ where $g(x) = 2x-1$, $f(x) = \sqrt{x}$.

Circles in Standard Form

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

• Circle at Origin: $x^2 + y^2 = r^2$

• Circle Not at Origin: $(x-h)^2 + (y-k)^2 = r^2$

• To graph, plot the center and use the radius to draw the circle.

• A circle is not a function because it fails the vertical line test.

Example: $(x-2)^2 + (y+3)^2 = 16$ is a circle centered at (2, -3) with radius 4.

General Form to Standard Form for Circles

The general form of a circle is $x^2 + y^2 + Dx + Ey + F = 0$. To convert to standard form, complete the square for x and y.

• Group x and y terms, complete the square, and rewrite in standard form.

Example: $x^2 + y^2 - 2x + 6y = 0$ becomes $(x-1)^2 + (y+3)^2 = 10$ after completing the square.

Practice and Application

• Identify functions from graphs and equations.

• Find domains and ranges using interval notation.

• Apply transformations to basic functions and analyze their effects.

• Perform operations and compositions on functions, and determine resulting domains.

• Work with equations of circles, graph them, and convert between general and standard forms.

Additional info: These notes are based on "Blitzer - Algebra and Trigonometry, Ch. 2: Functions and Graphs" and are suitable for Precalculus students preparing for exams or reviewing key concepts.