A5-A7

Functions and Their Properties

Identifying Features of Functions

Understanding the basic features of functions is essential in College Algebra. These features can be determined from equations, tables, or graphs.

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

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

• x-intercept: The point(s) where the graph crosses the x-axis (set y = 0).

• y-intercept: The point where the graph crosses the y-axis (set x = 0).

• Intervals of Increase/Decrease: Intervals where the function values are rising or falling as x increases.

• Intervals of Positivity/Negativity: Intervals where the function values are above or below the x-axis.

Example: For the function graphed below, the domain is $(-\infty, 7]$, the range is $[-2, 6]$, the x-intercepts are at $x = 0, 2, 6$, and the y-intercept is at $y = 2$.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• To analyze, create a table of values for each interval and graph each piece accordingly.

• Identify domain, range, intercepts, and intervals of increase/decrease for the entire function.

Example: $f(x) = \begin{cases} 2x+3 & x < -2 \\ -x+2 & x \geq -2 \end{cases}$

Operations on Functions

Adding, Subtracting, Multiplying, and Dividing Functions

Given two functions $f(x)$ and $g(x)$, you can combine them using the following operations:

• Addition: $(f+g)(x) = f(x) + g(x)$

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

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

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

The domain of the resulting function is the intersection of the domains of $f$ and $g$, except for division, where $g(x) \neq 0$.

Examples Using Tables, Equations, and Graphs

• Evaluate combined functions at specific values using tables or equations.

• For example, if $f(2) = 5$ and $g(2) = 3$, then $(f+g)(2) = 8$.

Composition of Functions

Definition and Notation

The composition of two functions $f$ and $g$ is written as $(f \circ g)(x) = f(g(x))$. This means you first apply $g$ to $x$, then apply $f$ to the result.

• Order matters: $(f \circ g)(x) \neq (g \circ f)(x)$ in general.

• Composition can be evaluated using tables, equations, or graphs.

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(2) = f(g(2)) = f(3) = 9$.

Applications of Function Operations and Composition

Real-World Applications

• Cost, revenue, and profit functions can be combined using addition, subtraction, or composition.

• For example, if $C(x)$ is the cost to produce $x$ items and $R(x)$ is the revenue, then profit is $P(x) = R(x) - C(x)$.

• Composition can model situations where one process depends on the output of another (e.g., population growth over time).

Inverse Functions

Definition and Properties

An inverse function $f^{-1}(x)$ reverses the effect of the original function $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$.

• A function has an inverse if and only if it is one-to-one (passes the Horizontal Line Test).

• The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y = x$.

Finding the Inverse Algebraically

1. Replace $f(x)$ with $y$.

2. Interchange $x$ and $y$.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$.

Example: Find the inverse of $f(x) = 2x+1$.

• Let $y = 2x+1$

• Switch $x$ and $y$: $x = 2y+1$

• Solve for $y$: $y = \frac{x-1}{2}$

• So $f^{-1}(x) = \frac{x-1}{2}$

Graphical Interpretation

• The inverse function undoes the action of the original function.

• Not all functions have inverses; the function must be one-to-one.

• Use the Horizontal Line Test to determine if a function is invertible.

Examples and Practice

• Given a table or graph, determine if the function is invertible and, if so, find the inverse values.

• Practice finding inverses for linear and rational functions.

Summary Table: Function Operations

Operation	Notation	Formula	Domain
Addition	$(f+g)(x)$	$f(x) + g(x)$	Intersection of domains of $f$ and $g$
Subtraction	$(f-g)(x)$	$f(x) - g(x)$	Intersection of domains of $f$ and $g$
Multiplication	$(fg)(x)$	$f(x) \cdot g(x)$	Intersection of domains of $f$ and $g$
Division	$\left(\frac{f}{g}\right)(x)$	$\frac{f(x)}{g(x)}$	Intersection of domains of $f$ and $g$, $g(x) \neq 0$
Composition	$(f \circ g)(x)$	$f(g(x))$	Domain of $g$ such that $g(x)$ is in domain of $f$

Key Takeaways

• Identify and interpret the domain, range, intercepts, and intervals of increase/decrease for functions from equations, tables, or graphs.

• Combine functions using addition, subtraction, multiplication, division, and composition, being careful with domains.

• Inverse functions reverse the input-output process; only one-to-one functions have inverses.

• Use algebraic and graphical methods to find and verify inverses.

Additional info: These notes include examples and applications relevant to College Algebra, such as cost/revenue functions and graphical analysis of invertibility.