Chapter 1.5, Problem 51E

(a)

To determine

To calculate: To determine the domain of the function $f$

Answer to Problem 51E

Domain is set of all real numbers

Explanation of Solution

Given: $f(x)=\left|x-4\right|$

Formula Used:

Domain of a function is the set of all possible inputs for the function

Calculation:

Function is given as follows:

  $f(x)=\left|x-4\right|$

Domain of $f\left(x\right)$ is set of all real numbers

Conclusion:

Hence, domain is set of all real numbers

(b)

To determine

To calculate: To determine the domain of the function $g$

Answer to Problem 51E

Domain is set of all real numbers

Explanation of Solution

Given: $g\left(x\right)=3-x$

Formula Used:

Domain of a function is the set of all possible inputs for the function

Calculation:

Function is given as follows:

  $g\left(x\right)=3-x$

Domain of $g\left(x\right)$ is set of all real numbers

Conclusion:

Hence, domain is set of all real numbers

(c)

To determine

To calculate: To determine the domain of the $f\circ g$

Answer to Problem 51E

Domain is set of all real numbers

Explanation of Solution

Given: $f(x)=\left|x-4\right|$ and $g\left(x\right)=3-x$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows:

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Also, Domain of a function is the set of all possible inputs for the function

Calculation:

Given function as follows:

  $f(x)=\left|x-4\right|$ and $g\left(x\right)=3-x$

  $f\circ g$ is calculated as follows:

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

Substitute $g\left(x\right)=3-x$

  $\left(f\circ g\right)\left(x\right)=f\left(3-x\right)$

Now substitute $f(x)=\left|x-4\right|$

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=\left|\left(3-x\right)-4\right|\\ \left(f\circ g\right)\left(x\right)=\left|3-x-4\right|\\ \left(f\circ g\right)\left(x\right)=\left|-1-x\right|\end{array}$

Now, Domain of $f\circ g$ is set of all real numbers

Conclusion:

Hence, Domain is set of all real numbers