2.7 Combining Functions

2.7.1 Learning Objectives

By the end of this section, you should be able to:

add, subtract, multiply, and divide functions;
determine the domain of a combined function;
calculate and interpret a composition of functions;
determine the domain of a composite function; and
decompose a composite function into simpler functions.

2.7.2 Arithmetic Combinations of Functions

Functions can be combined using ordinary arithmetic:

\[ \begin{aligned} (f+g)(x)&=f(x)+g(x),\\ (f-g)(x)&=f(x)-g(x),\\ (fg)(x)&=f(x)g(x),\\ \left(\frac{f}{g}\right)(x)&=\frac{f(x)}{g(x)},\qquad g(x)\ne0. \end{aligned} \]

Example 2.33 Combining Two Functions

Let $f(x)=x^2+1$ and $g(x)=3x-2$. Find $f+g$, $fg$, and $f/g$.

Solution

\[ \begin{aligned} (f+g)(x)&=x^2+3x-1,\\ (fg)(x)&=(x^2+1)(3x-2),\\ \left(\frac{f}{g}\right)(x)&=\frac{x^2+1}{3x-2},\qquad x\ne\frac23. \end{aligned} \]

\[ \boxed{(f+g)(x)=x^2+3x-1,\quad (fg)(x)=(x^2+1)(3x-2),\quad \left(\frac fg\right)(x)=\frac{x^2+1}{3x-2}} \]

Exercise 2.32 If $f(x)=2x+5$ and $g(x)=x-4$, find $(f-g)(x)$.

2.7.3 Domains of Combined Functions

For sums, differences, and products, use inputs common to the domains of both functions. For a quotient, also exclude inputs that make the denominator zero.

Example 2.34 Finding a Quotient’s Domain

Let $f(x)=\sqrt{x+1}$ and $g(x)=x-3$. Find the domain of $f/g$.

Solution

The square root requires $x\ge-1$, and the denominator requires $x\ne3$. Thus,

\[ \boxed{[-1,3)\cup(3,\infty)}. \]

Exercise 2.33 Find the domain of $\dfrac{\sqrt{x-2}}{x-5}$.

2.7.4 Composition of Functions

The composition of $f$ with $g$ is

\[ (f\circ g)(x)=f(g(x)). \]

Apply the inner function first. In general, $f\circ g$ and $g\circ f$ are different.

Example 2.35 Comparing Two Compositions

For $f(x)=x^2+1$ and $g(x)=3x-2$, find $f\circ g$ and $g\circ f$.

Solution

\[ \begin{aligned} (f\circ g)(x)&=(3x-2)^2+1=9x^2-12x+5,\\ (g\circ f)(x)&=3(x^2+1)-2=3x^2+1. \end{aligned} \]

\[ \boxed{(f\circ g)(x)=9x^2-12x+5,\quad (g\circ f)(x)=3x^2+1} \]

Exercise 2.34 If $f(x)=\sqrt{x}$ and $g(x)=x+6$, find $(f\circ g)(x)$.

2.7.5 Domains of Composite Functions

For $f\circ g$, an input $x$ must be in the domain of $g$, and the output $g(x)$ must be in the domain of $f$.

Example 2.36 Restricting a Composite Domain

Let $f(x)=\sqrt{x}$ and $g(x)=x-4$. Find $f\circ g$ and its domain.

Solution

\[ (f\circ g)(x)=\sqrt{x-4}. \]

The radicand must be non-negative:

\[ x-4\ge0. \]

Therefore,

\[ \boxed{(f\circ g)(x)=\sqrt{x-4},\quad \text{domain }[4,\infty)}. \]

Exercise 2.35 Let $f(x)=1/x$ and $g(x)=x^2-9$. Find the domain of $f\circ g$.

2.7.6 Decomposing a Composition

A complicated expression can often be viewed as an outer function applied to an inner function. This helps reveal its structure.

Example 2.37 Identifying Inner and Outer Functions

Express $H(x)=\sqrt{5x+1}$ as $f\circ g$.

Solution

Choose the inner function $g(x)=5x+1$ and the outer function $f(u)=\sqrt{u}$. Then

\[ f(g(x))=\sqrt{5x+1}. \]

\[ \boxed{f(u)=\sqrt u,\quad g(x)=5x+1} \]

Exercise 2.36 Express $H(x)=(2x-3)^4$ as $f\circ g$.

2.7.7 Applications and Iteration

Composition describes processes performed in sequence. Repeated composition of a function with itself is called iteration. The notation $f^2(x)$ may mean $f(f(x))$ in this context; it does not mean $[f(x)]^2$.

Example 2.38 Converting and Applying a Rate

The function $C(F)=\frac59(F-32)$ converts Fahrenheit to Celsius. A second function $W(C)=C-2$ models a wind adjustment. Find $W(C(68))$.

Solution

\[ C(68)=\frac59(68-32)=20, \]

and

\[ W(20)=18. \]

\[ \boxed{W(C(68))=18^\circ\text{C}} \]

Exercise 2.37 If $f(x)=0.5x+3$, calculate $f(f(10))$.

2.7.8 Conceptual Takeaways

Arithmetic combinations operate on the outputs of functions at the same input.
The domain of a combination must respect every restriction in the original functions.
Composition applies one complete function and then uses its output as the next input.
The order of composition matters.
Decomposition and iteration reveal how multi-stage processes are organized.

2.7.9 Skills You Should Be Able to Do

Calculate sums, differences, products, and quotients of functions.
Determine the domain of an arithmetic combination.
Evaluate and simplify composite functions.
Determine the domain of a composition.
Decompose and interpret multi-stage functions.

2.7.10 Practice Problems with Solutions

Let $f(x)=x+2$ and $g(x)=x^2$. Find $(f+g)(x)$.

Show Solution

\[ (f+g)(x)=x+2+x^2. \]

Thus, $\boxed{x^2+x+2}$.
For the functions in Problem 1, find $(fg)(-2)$.

Show Solution

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

\[ \boxed{(fg)(-2)=0(4)=0}. \]
Find the domain of $\dfrac{\sqrt{x+4}}{x-1}$.

Show Solution

The square root requires $x\ge-4$, and the denominator requires $x\ne1$. Hence,

\[ \boxed{[-4,1)\cup(1,\infty)}. \]
If $f(x)=2x-1$ and $g(x)=x^2+3$, find $(f\circ g)(x)$.

Show Solution

\[ f(g(x))=2(x^2+3)-1=2x^2+5. \]

Thus, $\boxed{(f\circ g)(x)=2x^2+5}$.
For the functions in Problem 4, find $(g\circ f)(x)$.

Show Solution

\[ g(f(x))=(2x-1)^2+3=4x^2-4x+4. \]

Thus, $\boxed{(g\circ f)(x)=4x^2-4x+4}$.
Let $f(x)=\sqrt{x+1}$ and $g(x)=2x-5$. Find $f\circ g$ and its domain.

Show Solution

\[ (f\circ g)(x)=\sqrt{(2x-5)+1}=\sqrt{2x-4}. \]

The radicand requires $2x-4\ge0$, so

\[ \boxed{(f\circ g)(x)=\sqrt{2x-4},\quad \text{domain }[2,\infty)}. \]
Decompose $H(x)=\dfrac{1}{(x+4)^2}$ as $f\circ g$.

Show Solution

Let $g(x)=(x+4)^2$ and $f(u)=1/u$. Then $f(g(x))=1/(x+4)^2$.

\[ \boxed{g(x)=(x+4)^2,\quad f(u)=\frac1u} \]
A price $p$ is increased by $8\%$, then a $\$5$ delivery fee is added. Write the final cost as a composition and find the cost when $p=\$75$.

Show Solution

Let $I(p)=1.08p$ apply the increase and $D(x)=x+5$ add delivery. Then

\[ (D\circ I)(p)=1.08p+5. \]

For $p=75$,

\[ (D\circ I)(75)=1.08(75)+5=86. \]

Therefore, $\boxed{\$86}$.