Study Guide: Relations, Functions, and Graphs in Precalculus

Relations and Functions

Definitions and Examples

Understanding the concepts of relations and functions is fundamental in precalculus. A relation is any set of ordered pairs (x, y), while a function is a special type of relation where each input x has exactly one output y.

• Relation: Any set of ordered pairs (x, y).

• Function: Each x corresponds to exactly one y.

• Example of a Function: {(1, 4), (2, 5), (3, 6)} — each x has one output.

• Non-Example: {(1, 4), (1, 7), (2, 5)} — x = 1 has two outputs; not a function.

Vertical Line Test

The vertical line test is a graphical method to determine if a relation is a function. If every vertical line intersects the graph at most once, the graph represents a function.

• Two or more intersections with a vertical line indicate the graph is not a function.

Function Evaluation

Substitution and Examples

To evaluate a function, substitute the given value for every x in the function's expression.

• To find $f(a)$, replace every x with a.

• For $f(x + h)$, replace every x with $(x + h)$.

• Example: If $f(x) = 2x^2 - 3x + 1$, then:

$\begin{align*} f(2) &= 2(2)^2 - 3(2) + 1 \\ &= 8 - 6 + 1 \\ &= 3 \end{align*}$

• Tip: Use parentheses around negative values to avoid errors.

Domain and Range

Definitions and Interval Notation

The domain of a function is the set of all possible x-values, and the range is the set of all corresponding y-values. Interval notation is used to describe these sets.

• Interval Notation:

  • (a, b): $a < x < b$

  • [a, b]: $a \leq x \leq b$

  • (a, b]: $a < x \leq b$

  • [a, b): $a \leq x < b$

  • (-\infty, a): $x < a$

  • [a, \infty): $x \geq a$

• Open circle: endpoint is not included.

• Closed circle: endpoint is included.

Domain Rules

• Denominator: Cannot be zero.

• Even-root: Radicand must be greater than or equal to zero.

Example 1: $f(x) = \sqrt{x - 3}$ Requires $x - 3 \geq 0$ so $x \geq 3$ Domain: $[3, \infty)$

Example 2: $g(x) = \frac{2}{x + 5}$ Requires $x + 5 \neq 0$ so $x \neq -5$ Domain: $(-\infty, -5) \cup (-5, \infty)$

Intercepts

x-intercept and y-intercept

Intercepts are points where the graph crosses the axes.

• x-intercept: $(x, 0)$ — set y = 0 and solve for x.

• y-intercept: $(0, y)$ — set x = 0 and solve for y.

• A graph can have zero, one, or multiple x-intercepts, but at most one y-intercept.

Example: $f(x) = 2 - 2x$ Find x-intercept: Set $y = 0$ $0 = 2 - 2x \Rightarrow x = 1$ Find y-intercept: Set $x = 0$ $y = 2 - 2(0) = 2$ So, x-intercept: $(1, 0)$, y-intercept: $(0, 2)$

Even, Odd, and Neither Functions

Definitions and Symmetry

Functions can be classified based on their symmetry properties.

• Even Function: $f(-x) = f(x)$ — symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ — symmetric about the origin.

• Neither: If neither condition is met.

Example (Even): $f(x) = x^2$ $f(-x) = (-x)^2 = x^2$

Example (Odd): $f(x) = x^3$ $f(-x) = -x^3 = -f(x)$

Example (Neither): $f(x) = x^2 + x$ $f(-x) = x^2 - x$

Increasing, Decreasing, and Constant Functions

Behavior of Functions

The behavior of a function can be described as increasing, decreasing, or constant over intervals.

• Increasing: Graph goes up as x increases.

• Decreasing: Graph goes down as x increases.

• Constant: Graph remains horizontal as x increases.

Example: $f(x) = (x - 3)^2$ Decreasing on $(-\infty, 3)$, increasing on $(3, \infty)$

Minima and Maxima

Local and Absolute Extrema

Functions can have local and absolute maximum and minimum values.

• Local Maximum: A nearby peak.

• Local Minimum: A nearby valley.

• Absolute Maximum: Highest y-value on the domain.

• Absolute Minimum: Lowest y-value on the domain.

Example: $f(x) = |x - 4|$ Local and absolute minimum at $(4, 0)$; no local or absolute maximum.

Basic Function Shapes

Common Functions

Recognizing basic function shapes is important for graphing and analysis.

• Linear: $f(x) = x$

• Quadratic: $f(x) = x^2$

• Absolute Value: $f(x) = |x|$

• Square Root: $f(x) = \sqrt{x}$

Function Transformations

Types of Transformations

Functions can be transformed by shifting, stretching, compressing, or reflecting their graphs.

Equation	Transformation
$f(x) + k$	Up k units
$f(x) - k$	Down k units
$f(x - h)$	Right h units
$f(x + h)$	Left h units
$a f(x)$	Vertical stretch/compression
$-f(x)$	Reflect over x-axis
$f(-x)$	Reflect over y-axis

Example: The graph of $y = \sqrt{x}$, shifted 3 units left, reflected across the x-axis, and shifted 5 units up: Start with $y = \sqrt{x}$ Shift left: $y = \sqrt{x + 3}$ Reflect x-axis: $y = -\sqrt{x + 3}$ Shift up: $y = -\sqrt{x + 3} + 5$

Practice Problems

• Find the domain of $g(x) = \sqrt{10 - x}$

• Check for symmetry: $f(x) = -2x^3 + 3x$

• Graph a function that is increasing on $(-\infty, -10)$, has a local maximum at $x = -10$, an absolute maximum at $(2, 5)$, and exactly 3 x-intercepts.

• Find x and y intercepts of $y = 8 + 5x$

• Write the equation for the graph of $y = \sqrt{x}$ shifted 3 units left, reflected across the x-axis, and shifted 5 units up.