A5-A7

Functions: Features and Operations

Identifying Features of Functions

Understanding the features of a function is essential in College Algebra. These features can be identified from graphs, tables, or equations and include domain, range, intercepts, and intervals of increase/decrease.

• Domain: The set of all possible input values (x-values) for which the function is defined. Example: For $f(x)$ shown on a graph, domain might be $(- rac{1}{2}, 7]$.

• Range: The set of all possible output values (y-values) the function can produce. Example: Range could be $[-2, 6]$.

• x-intercept: The point(s) where the graph crosses the x-axis (i.e., $f(x) = 0$).

• y-intercept: The point where the graph crosses the y-axis (i.e., $x = 0$).

• Intervals of Increase/Decrease:

  • Increasing: Where the function rises as x increases.

  • Decreasing: Where the function falls as x increases.

• Intervals of Positivity/Negativity:

  • Positive: Where $f(x) > 0$.

  • Negative: Where $f(x) < 0$.

Example Table: Features of a Graph

Feature	Example Value
Domain	$(- rac{1}{2}, 7]$
Range	$[-2, 6]$
x-intercept	$(2, 0)$
y-intercept	$(0, 4)$
Increasing Interval	$(0, 4)$
Decreasing Interval	$(- rac{1}{2}, 0)$ and $(4, 7]$

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, and division. These operations can be performed using tables, equations, or graphs.

• Addition: $(f+g)(x) = f(x) + g(x)$

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

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

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

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

Composition of Functions

Composition involves using the output of one function as the input of another. The notation is $(f \circ g)(x) = f(g(x))$.

• Order matters: $f(g(x))$ is generally not the same as $g(f(x))$.

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

Evaluating Operations and Compositions

Operations and compositions can be evaluated using tables, equations, or graphs.

• Using Tables: Substitute values from the table into the operation or composition.

• Using Equations: Substitute the value of $x$ into the function equations and perform the operation.

• Using Graphs: Read the value of $f(x)$ or $g(x)$ from the graph and perform the operation.

Applications of Function Operations

Function operations are used in real-world applications, such as modeling costs, revenues, and other quantities.

• Example: If $C(x)$ is the cost to make $x$ units and $R(x)$ is the revenue from selling $x$ units, then profit $P(x) = R(x) - C(x)$.

Inverse Functions

Definition and Properties

An inverse function $f^{-1}(x)$ reverses the roles of input and output. If $f(a) = b$, then $f^{-1}(b) = a$.

• One-to-one function: A function must be one-to-one (pass the horizontal line test) to have an inverse.

• Finding the inverse:

  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 $y = 2x + 1$

• Interchange $x$ and $y$: $x = 2y + 1$

• Solve for $y$: $y = \frac{x - 1}{2}$

• So, $f^{-1}(x) = \frac{x - 1}{2}$

Graphical Interpretation

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

• Domain of $f$ becomes range of $f^{-1}$ and vice versa.

Testing for Invertibility

• If a function fails the horizontal line test, it does not have an inverse.

• Not all functions are invertible.

Practice Problems and Applications

• Given a table, graph, or equation, determine if a function is invertible and find its inverse if possible.

• Apply inverse functions to solve real-world problems, such as finding the original value given an output.

Summary Table: Operations and Inverses

Operation	Formula	Domain Condition
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	$(fg)(x) = f(x)g(x)$	Intersection of domains of $f$ and $g$
Division	$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$	Intersection of domains, $g(x) \neq 0$
Composition	$(f \circ g)(x) = f(g(x))$	$x$ in domain of $g$, $g(x)$ in domain of $f$
Inverse	$f^{-1}(x)$	$f$ must be one-to-one

Key Takeaways

• Identify domain, range, intercepts, and intervals from graphs, tables, or equations.

• Perform operations and compositions on functions in all representations.

• Determine if a function is invertible and find its inverse using algebraic and graphical methods.

• Apply these concepts to solve real-world problems.

Additional info: These notes cover topics from College Algebra chapters on Functions, Operations on Functions, and Inverse Functions, including graphical, tabular, and algebraic representations.