Chapter 2.6, Problem 4E

We can express the functions in Exercise 3 algebraically as

f(x) = _____

g(x) = _____

(f ○ g)(x) = _____

(g ○ f)(x) = _____

3. If the rule of the function f is “add one” and the rule of the function g is “multiply by 2,” then the rule of f ○ g is “____________________”, and the rule of g ○ f is “____________________.”

To determine

To evaluate: The values of functions $f\left(x\right)$, $g\left(x\right)$, $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$ if the rule of the function f is add one and the rule of the function g is multiply by 2.

Answer to Problem 4E

The values of function $f\left(x\right)$ is $g\left(x\right)$ is $2x$, $\left(f\circ g\right)\left(x\right)$ is $2x+1$, $\left(g\circ f\right)\left(x\right)$ is $2\left(x+1\right)$.

Explanation of Solution

Section1:

The rule of the function f is adding one.

So, the value of function $f\left(x\right)$ is added by one.

$f\left(x\right)=x+1$

Thus, the value of function $f\left(x\right)$ is $x+1$.

Section2:

The rule of the function f is multiply by 2.

So, the value of function $g\left(x\right)$ is multiplied by one.

$g\left(x\right)=2x$

Thus, the value of function $g\left(x\right)$ is $2x$.

Section3:

The rule of the function f is add one and the rule of the function g is multiply by 2.

From section 2:

The value of function $f\left(x\right)$ is $x+1$.

From section3:

The value of function $g\left(x\right)$ is $2x$.

The function $f\circ g$ is called the compositions of functions f and g and the domain of $f\circ g$ is set of all x in the domain of g such that $g\left(x\right)$ is in the domain of f, that means,

$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Substitute the values of $f\left(x\right)$ and $g\left(x\right)$ in above formula,

$\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(2x\right)\\ =2x+1\end{array}$

Thus, the value of function $\left(f\circ g\right)\left(x\right)$ is $2x+1$.

Section4:

The rule of the function f is add one and the rule of the function g is multiply by 2.

From section 2:

The value of function $f\left(x\right)$ is $x+1$.

From section3:

The value of function $g\left(x\right)$ is $2x$.

The function $g\circ f$ is called the compositions of functions g and f and the domain of $g\circ f$ is set of all x in the domain of f such that $f\left(x\right)$ is in the domain of g, that means,

$\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

Substitute the values of $f\left(x\right)$ and $g\left(x\right)$ in above formula,

$\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left(x+1\right)\\ =2\left(x+1\right)\end{array}$

Thus, the value of function $\left(g\circ f\right)\left(x\right)$ is $2\left(x+1\right)$.