Chapter 1, Problem 20CT

(a)

To determine

To calculate: The expression of the function $\left(f+g\right)\left(x\right)$

Answer to Problem 20CT

The expression of the function $\left(f+g\right)\left(x\right)$ is $\frac{1+2x^{\frac{3}{2}}}{x}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=\frac{1}{x}$

And,

  $g\left(x\right)=2\sqrt{x}$

Calculation:

Find the expression of the function $\left(f+g\right)\left(x\right)$ ,

We can write as,

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

Substitute the function $f\left(x\right)\text{ and }g\left(x\right)$ in the above expression,

  $\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =\frac{1}{x}+2\sqrt{x}\\ =\frac{1+2x^{\frac{3}{2}}}{x}\end{array}$

Simplifying further,

  $\left(f+g\right)\left(x\right)=\frac{1+2x^{\frac{3}{2}}}{x}$

Hence, the expression of the function $\left(f+g\right)\left(x\right)$ is $\frac{1+2x^{\frac{3}{2}}}{x}$ .

(b)

To determine

To calculate: The expression of the function $\left(f-g\right)\left(x\right)$

Answer to Problem 20CT

The expression of the function $\left(f-g\right)\left(x\right)$ is $\frac{1-2x^{\frac{3}{2}}}{x}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=\frac{1}{x}$

And,

  $g\left(x\right)=2\sqrt{x}$

Calculation:

Find the expression of the function $\left(f-g\right)\left(x\right)$ ,

We can write as,

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

Substitute the function $f\left(x\right)\text{ and }g\left(x\right)$ in the above expression,

  $\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =\frac{1}{x}-2\sqrt{x}\\ =\frac{1-2x^{\frac{3}{2}}}{x}\end{array}$

Simplifying further,

  $\left(f-g\right)\left(x\right)=\frac{1-2x^{\frac{3}{2}}}{x}$

Hence, the expression of the function $\left(f-g\right)\left(x\right)$ is $\frac{1-2x^{\frac{3}{2}}}{x}$ .

(c)

To determine

To calculate: The expression of the function $\left(fg\right)\left(x\right)$

Answer to Problem 20CT

The expression of the function $\left(fg\right)\left(x\right)$ is $\frac{2}{\sqrt{x}}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=\frac{1}{x}$

And,

  $g\left(x\right)=2\sqrt{x}$

Calculation:

Find the expression of the function $\left(fg\right)\left(x\right)$ ,

We can write as,

  $\left(fg\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$

Substitute the function $f\left(x\right)\text{ and }g\left(x\right)$ in the above expression,

  $\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\\ =\frac{1}{x}\cdot 2\sqrt{x}\\ =\frac{2\sqrt{x}}{x}\\ =\frac{2}{\sqrt{x}}\end{array}$

Simplifying further,

  $\left(fg\right)\left(x\right)=\frac{2}{\sqrt{x}}$

Hence, theexpression of the function $\left(fg\right)\left(x\right)$ is $\frac{2}{\sqrt{x}}$ .

(d)

To determine

To calculate: The expression of the function $\left(\frac{f}{g}\right)\left(x\right)$

Answer to Problem 20CT

The expression of the function $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{1}{2x^{\frac{3}{2}}}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=\frac{1}{x}$

And,

  $g\left(x\right)=2\sqrt{x}$

Calculation:

Find theexpression of thefunction $\left(\frac{f}{g}\right)\left(x\right)$ ,

We can write as,

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

Substitute the function $f\left(x\right)\text{ and }g\left(x\right)$ in the above expression,

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{\frac{1}{x}}{2\sqrt{x}}\\ =\frac{1}{2\sqrt{x}\cdot x}\\ =\frac{1}{2x^{\frac{3}{2}}}\end{array}$

Simplifying further,

  $\left(\frac{f}{g}\right)\left(x\right)=\frac{1}{2x^{\frac{3}{2}}}$

Hence, the expression of the function $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{1}{2x^{\frac{3}{2}}}$ .

(e)

To determine

To calculate: The expression of the function $\left(f\circ g\right)\left(x\right)$

Answer to Problem 20CT

The expression of the function $\left(f\circ g\right)\left(x\right)$ is $\frac{2}{\sqrt{x}}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=\frac{1}{x}$

And,

  $g\left(x\right)=2\sqrt{x}$

Calculation:

Find the expression of the function $\left(f\circ g\right)\left(x\right)$ ,

We can write as,

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

Substitute the function $f\left(x\right)\text{ and }g\left(x\right)$ in the above expression,

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

Simplifying further,

  $\left(f\circ g\right)\left(x\right)=\frac{2}{\sqrt{x}}$

Hence, the expression of the function $\left(f\circ g\right)\left(x\right)$ is $\frac{2}{\sqrt{x}}$ .

(f)

To determine

To calculate: The expression of the function $\left(g\circ f\right)\left(x\right)$

Answer to Problem 20CT

The expression of the function $\left(g\circ f\right)\left(x\right)$ is $\frac{1}{2\sqrt{x}}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=\frac{1}{x}$

And,

  $g\left(x\right)=2\sqrt{x}$

Calculation:

Find the expression of the function $\left(g\circ f\right)\left(x\right)$ ,

We can write as,

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

Substitute the function $f\left(x\right)\text{ and }g\left(x\right)$ in the above expression,

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

Simplifying further,

  $\left(g\circ f\right)\left(x\right)=\frac{1}{2\sqrt{x}}$

Hence, the expression of the function $\left(g\circ f\right)\left(x\right)$ is $\frac{1}{2\sqrt{x}}$ .