Combining and Composing Functions in Precalculus

Chapter 1: Graphs and Functions

1.8 Combining Functions; Composite Functions

This section explores how to combine functions using basic operations and how to form and analyze composite functions. Understanding these concepts is essential for modeling and solving real-world problems in mathematics and science.

Basic Operations on Functions

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

The domain of each operation is the set of all $x$ values common to the domains of $f$ and $g$, with the additional restriction for the quotient that $g(x) \neq 0$.

Example: Combining Functions

Let $f(x) = x^2 - 6x + 8$ and $g(x) = x - 2$. Find the following:

• (f - g)(4): $f(4) = 0$, $g(4) = 2$, so $(f - g)(4) = 0 - 2 = -2$

• (f + g)(x): $(x^2 - 6x + 8) + (x - 2) = x^2 - 5x + 6$

• (f - g)(x): $(x^2 - 6x + 8) - (x - 2) = x^2 - 7x + 10$

• (fg)(x): $(x^2 - 6x + 8)(x - 2) = x^3 - 8x^2 + 20x - 16$

Finding the Domain of Combined Functions

To find the domain of $f + g$, $f - g$, $fg$, and $\frac{f}{g}$, determine the intersection of the domains of $f$ and $g$, excluding values where $g(x) = 0$ for the quotient.

For example, if $f(x)$ is defined for $x \geq -2$ and $g(x)$ for $x \leq 5$, then the domain of their combination is $[-2, 5]$.

Composite Functions

The composition of $f$ with $g$ is written as $(f \circ g)(x) = f(g(x))$. The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Example: Evaluating Composite Functions

• Let $f(x) = x^3$ and $g(x) = x + 1$.

• $(f \circ g)(2) = f(g(2)) = f(3) = 27$

• $(g \circ f)(2) = g(f(2)) = g(8) = 9$

Finding the Domain of a Composite Function

To find the domain of $f \circ g$:

1. Find the domain of $g$ (call it $A$).

2. Find the set $B$ of $x$ in $A$ for which $g(x)$ is in the domain of $f$.

3. The domain of $f \circ g$ is $B$.

For example, if $g(x)$ is defined for $x \leq 4$ and $f(x)$ is defined for $x \geq 0$, then $f \circ g$ is defined for those $x \leq 4$ such that $g(x) \geq 0$.

Decomposing a Function

Given a function $H(x)$, you can write it as a composition $f(g(x))$ by identifying an inner function $g(x)$ and an outer function $f(x)$ such that $H(x) = f(g(x))$.

• Example: $H(x) = \sqrt{2x^2 + 1}$ can be written as $f(g(x))$ where $g(x) = 2x^2 + 1$ and $f(x) = \sqrt{x}$.

Applications of Composition

• Area of an Oil Spill: If the radius $r$ of a circular oil slick increases at 2 miles per hour, then $r(t) = 2t$ and the area $A(t) = \pi [r(t)]^2 = \pi (2t)^2 = 4\pi t^2$. After 6 hours, $A(6) = 144\pi$ square miles.

• Car Pricing: If a car's MSRP is $x$, a rebate is $r(x) = x - 4000$, and a dealer discount is $d(x) = 0.92x$. The order of applying these functions affects the final price. For example, $d(r(x)) = 0.92(x - 4000)$ and $r(d(x)) = 0.92x - 4000$. Applying the rebate first and then the discount costs $320$ more than applying the discount first and then the rebate.

Summary Table: Operations on Functions

Operation	Definition	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: The examples and applications provided illustrate how combining and composing functions are used in practical scenarios, such as calculating areas and determining pricing strategies.