Combining and Composing Functions: Operations, Domains, and Applications

Combining Functions

Operations on Functions

When two functions f(x) and g(x) are both defined, we can create new functions by adding, subtracting, multiplying, or dividing them. These operations are defined as follows:

• Sum: $(f+g)(x) = f(x) + g(x)$

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

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

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

The domain of each combined function is the set of all real numbers for which both f and g are defined (and, for the quotient, where g(x) ≠ 0).

Examples and Applications

• Evaluating Combined Functions: To find $(f+g)(0)$, add the values $f(0)$ and $g(0)$.

• Tabular Representation: Functions can be represented in tables, where each input value corresponds to an output for f and g. Operations are performed entry-wise.

Domain Considerations

• Sum and Difference: The domain is the intersection of the domains of f and g.

• Product: The domain is also the intersection of the domains of f and g.

• Quotient: The domain is the intersection of the domains of f and g, excluding values where g(x) = 0.

Example: If $f(x) = \sqrt{x}$ and $g(x) = x-2$, then the domain of $\frac{f}{g}$ is $x \geq 0$ and $x \neq 2$.

Evaluating Functions Using Graphs and Tables

Graphical and Tabular Evaluation

Functions can be evaluated using their graphs or tables. For a given input, locate the corresponding output on the graph or in the table, then perform the required operation.

• Example: If $f(-3) = 5$ and $g(-3) = 7$, then $(f+g)(-3) = 5 + 7 = 12$.

• When using a graph, read the y-value for the given x-value.

Composition of Functions

Definition and Notation

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

• Formula: $(f \circ g)(x) = f(g(x))$

• Domain: The domain of $f \circ g$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

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

Evaluating Compositions with Tables and Graphs

• To evaluate $f(g(x))$ using a table, find $g(x)$ first, then use that output as the input for $f$.

• When using a graph, trace the output of $g(x)$ and use it as the input for $f$.

Domain of Composite Functions

Finding the Domain

To determine the domain of a composite function $f(g(x))$:

1. Find the domain of $g(x)$.

2. Find the domain of $f(x)$.

3. Restrict the domain of $g(x)$ to values for which $g(x)$ is in the domain of $f$.

Example: If $f(x) = \sqrt{x}$ and $g(x) = 2x-1$, then $f(g(x)) = \sqrt{2x-1}$, and the domain is $x \geq 0.5$.

Summary Table: Operations and Domains

Operation	Formula	Domain
Sum	$(f+g)(x) = f(x) + g(x)$	Intersection of domains of $f$ and $g$
Difference	$(f-g)(x) = f(x) - g(x)$	Intersection of domains of $f$ and $g$
Product	$(fg)(x) = f(x) \cdot g(x)$	Intersection of domains of $f$ and $g$
Quotient	$\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)}$	Intersection of domains of $f$ and $g$, $g(x) \neq 0$
Composition	$(f \circ g)(x) = f(g(x))$	All $x$ in domain of $g$ such that $g(x)$ is in domain of $f$

Additional info: These notes cover foundational skills for combining and composing functions, which are essential for understanding more advanced topics in algebra and calculus. Mastery of these concepts is critical for solving equations, modeling real-world scenarios, and analyzing mathematical relationships.