Precalculus Exam 1 Review: Functions, Graphs, and Applications

Functions and Their Representations

Definition and Interpretation of Functions

A function is a relation that assigns exactly one output value for each input value from a specified domain. Functions can be represented in various forms, including tables, graphs, formulas, and verbal descriptions.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually y-values).

• Function Notation: $f(x)$ denotes the output of function f for input x.

• Example: If $f(x)$ is the number of ducks in a lake x years after 1990, then $f(0)$ is the number of ducks in 1990.

Evaluating and Interpreting Functions

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

• Interpretation often involves understanding the context, such as population, height, or cost.

• Example: If $h(t)$ is the height above ground t seconds after launch, $h(2)$ gives the height at 2 seconds.

Types of Functions and Their Properties

Identifying Functions from Tables and Graphs

To determine if a relation is a function, check that each input corresponds to exactly one output.

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

• Table Representation: Each input value should appear only once.

Domain of Functions

The domain of a function is the set of all input values for which the function is defined.

• For rational functions, exclude values that make the denominator zero.

• For square root functions, the expression under the root must be non-negative.

• Examples:

  • $f(x) = \frac{1}{x-2}$ has domain $x \neq 2$.

  • $g(x) = \sqrt{x-1}$ has domain $x \geq 1$.

Piecewise and Composite Functions

Piecewise Functions

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

• To write a formula for a piecewise function, specify the formula and the interval for each piece.

• Example:

  • $f(x) = \begin{cases} -x & x \leq 1 \\ x^2 & 1 < x \leq 2 \\ 2 & x > 2 \end{cases}$

Composite Functions

The composition of two functions $f$ and $g$ is written as $f(g(x))$ and means to apply $g$ first, then $f$.

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

Graphing and Analyzing Functions

Graphing Functions

• Plot points by evaluating the function at various input values.

• For piecewise functions, plot each piece on its specified interval.

• Use the vertical line test to confirm the graph represents a function.

Average Rate of Change

The average rate of change of a function $f$ on the interval $[a, b]$ is given by:

$\frac{f(b) - f(a)}{b - a}$

• Represents the slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$.

• Example: For $f(x) = x^2$ on $[1, 3]$, the average rate of change is $\frac{9-1}{3-1} = 4$.

Transformations of Functions

Types of Transformations

• Vertical Shifts: $f(x) + k$ shifts the graph up by $k$ units.

• Horizontal Shifts: $f(x-h)$ shifts the graph right by $h$ units.

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

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

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

Examples of Transformations

• $f(x) = x^2$ shifted up 3 units: $f(x) + 3 = x^2 + 3$

• $f(x) = x^2$ reflected across the x-axis: $-x^2$

• $f(x) = x^2$ horizontally stretched by a factor of 2: $f(\frac{x}{2}) = (\frac{x}{2})^2$

Special Types of Functions

Even and Odd Functions

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

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

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

Linear Functions and Applications

Equations of Lines

• 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.

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals.

Applications of Linear Functions

• Modeling population growth, cost, revenue, and other real-world scenarios.

• Finding the intersection point of two lines can represent equilibrium or break-even points.

Tables

Sample Table: Determining if a Relation is a Function

x	y
1	2
2	3
3	4

This table represents a function because each x-value corresponds to exactly one y-value.

Summary

• Understand the definition and properties of functions.

• Be able to evaluate, graph, and transform functions.

• Apply concepts to real-world problems, including interpreting and constructing linear models.

• Recognize and work with piecewise, composite, even, and odd functions.