Chapter 1.5, Problem 52E

(a)

To determine

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

Answer to Problem 52E

Domain is $R-\left\{0\right\}$

Explanation of Solution

Given: $f\left(x\right)=\frac{2}{\left|x\right|}$

Formula Used:

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

Calculation:

Function is given as follows:

  $f\left(x\right)=\frac{2}{\left|x\right|}$

For $f\left(x\right)$ to be defined, $x\ne 0$

Domain of $f\left(x\right)$ is set of all real numbers except $0$

Thus, domain is $R-\left\{0\right\}$

Conclusion:

Hence, domain is $R-\left\{0\right\}$

(b)

To determine

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

Answer to Problem 52E

Domain is $R$

Explanation of Solution

Given: $g\left(x\right)=x-5$

Formula Used:

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

Calculation:

Function is given as follows:

  $g\left(x\right)=x-5$

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

Thus, domain is $R$

Conclusion:

Hence, domain is $R$

(c)

To determine

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

Answer to Problem 52E

Domain is $R-\left\{5\right\}$

Explanation of Solution

Given: $f\left(x\right)=\frac{2}{\left|x\right|}$ and $g\left(x\right)=x-5$

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:

  $f\left(x\right)=\frac{2}{\left|x\right|}$ and $g\left(x\right)=x-5$

  $f\circ g$ is calculated as follows:

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

Substitute $g\left(x\right)=x-5$

  $\left(f\circ g\right)\left(x\right)=f\left(x-5\right)$

Now substitute $f\left(x\right)=\frac{2}{\left|x\right|}$

  $\left(f\circ g\right)\left(x\right)=\frac{2}{\left|x-5\right|}$

For $\left(f\circ g\right)\left(x\right)$ to be defined, $x\ne 5$

Domain of $f\circ g$ is set of all real numbers expect $5$

Thus, domain is $R-\left\{5\right\}$

Conclusion:

Hence, Domain is $R-\left\{5\right\}$