College Algebra: Functions, Graphs, and Their Properties

Functions and Relations

Identifying Functions

A function is a relation in which each input (domain value) is paired with exactly one output (range value). To determine if a relation is a function, check that no input value corresponds to more than one output value.

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

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

• Example: For the set {(-4, 19), (3, 10), (0, 1), (3, 11), (5, 23)}, since the input 3 is paired with both 10 and 11, this relation is not a function.

Equations Defining y as a Function of x

To determine if an equation defines y as a function of x, solve for y and check if each x-value yields only one y-value.

• Example: $4x - x^2 = y + 24$ Rearranged: $y = 4x - x^2 - 24$ For each x, there is only one y, so this is a function.

Evaluating and Combining Functions

Function Evaluation

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

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

Combining Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Example: If $f(x) = 9x - 4$ and $g(x) = 2x + 6$, then:

  • $(f + g)(x) = f(x) + g(x) = (9x - 4) + (2x + 6) = 11x + 2$

  • $(f \cdot g)(x) = f(x) \cdot g(x) = (9x - 4)(2x + 6)$

Properties of Functions

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all x in the domain, and odd if $f(-x) = -f(x)$ for all x.

• Example: $f(x) = |x|$ is even because $|-x| = |x|$.

• Example: $f(x) = x^3$ is odd because $(-x)^3 = -x^3 = -f(x)$.

Graphing and Analyzing Functions

Graph Interpretation

Graphs can be used to determine the domain, range, intercepts, and symmetry of a function.

• Domain: The set of x-values for which the function is defined.

• Range: The set of y-values the function attains.

• Intercepts: Points where the graph crosses the axes (x-intercepts and y-intercepts).

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

Finding the Slope Between Two Points

The slope of the line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

• $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Between (2, 4) and (1, 9): $m = \frac{9 - 4}{1 - 2} = \frac{5}{-1} = -5$

Average Rate of Change

The average rate of change of a function $f$ from $x = a$ to $x = b$ is:

• $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Example: For $f(x) = 4x - 11$ from $x = 1$ to $x = 9$: $f(9) = 4 \times 9 - 11 = 36 - 11 = 25$ $f(1) = 4 \times 1 - 11 = 4 - 11 = -7$ Average rate: $\frac{25 - (-7)}{9 - 1} = \frac{32}{8} = 4$

Using Graphs to Analyze Functions

Graphs can be used to determine where a function is increasing, decreasing, or constant, and to find specific function values.

• Example: For the graph of $f(x) = -x^2 + 8x$ (see Figure 2):

  • The function increases on $(-\infty, 4)$ and decreases on $(4, \infty)$.

  • To find $x$ such that $f(x) = 8.25$, solve $-x^2 + 8x = 8.25$.

  • From the graph, $x \approx 0.5$ and $x \approx 7.5$ (actual values may vary based on the graph's accuracy).

Special Functions and Their Graphs

Absolute Value and Piecewise Functions

The absolute value function is defined as $f(x) = |x|$. Its graph is a "V" shape, symmetric about the y-axis.

• Piecewise functions can be defined using absolute value, such as $f(x) = |x| - 7$.

• Example: $f(x) = |x| - 7$ shifts the graph of $|x|$ downward by 7 units.

Quadratic Functions

A quadratic function has the form $f(x) = ax^2 + bx + c$. Its graph is a parabola.

• If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• The vertex is at $x = -\frac{b}{2a}$.

• Example: $f(x) = -x^2 + 8x$ has vertex at $x = 4$, $f(4) = -(4)^2 + 8 \times 4 = -16 + 32 = 16$.

Summary Table: Function Properties

Function	Even/Odd/Neither	Domain	Range	Symmetry
$f(x) = |x|$	Even	All real numbers	$[0, \infty)$	y-axis
$f(x) = x^3$	Odd	All real numbers	All real numbers	Origin
$f(x) = -x^2 + 8x$	Neither	All real numbers	Varies (see graph)	None

Additional info:

• Some function properties and graph interpretations were inferred based on standard College Algebra curriculum and the provided images.

• Specific graph points and values (e.g., $x$ for $f(x) = 8.25$) were estimated from the graph and may require precise calculation for exact answers.