Chapter 1, Problem 19CT

(a)

To determine

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

Answer to Problem 19CT

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

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=3x^2-7$

And,

  $g\left(x\right)=-x^2-4x+5$

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,

  $\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =\left(3x^2-7\right)+\left(-x^2-4x+5\right)\\ =3x^2-7-x^2-4x+5\\ =2x^2-4x-2\end{array}$

Simplifying further,

  $\left(f+g\right)\left(x\right)=2\left(x^2-2x-1\right)$

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

(b)

To determine

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

Answer to Problem 19CT

The expression of the function $\left(f-g\right)\left(x\right)$ is $4\left(x^2+x-3\right)$ .

Explanation of Solution

Calculation:

Given information:

Consider the given function,

  $f\left(x\right)=3x^2-7$

And,

  $g\left(x\right)=-x^2-4x+5$

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,

  $\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =\left(3x^2-7\right)-\left(-x^2-4x+5\right)\\ =3x^2-7+x^2+4x-5\\ =4x^2+4x-12\end{array}$

Simplifying further,

  $\left(f-g\right)\left(x\right)=4\left(x^2+x-3\right)$

Hence, the expression of the function $\left(f-g\right)\left(x\right)$ is $4\left(x^2+x-3\right)$ .

(c)

To determine

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

Answer to Problem 19CT

The expression of the function $\left(fg\right)\left(x\right)$ is $-3x^4-12x^3+22x^2+28x-35$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=3x^2-7$

And,

  $g\left(x\right)=-x^2-4x+5$

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,

  $\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\\ =\left(3x^2-7\right)\cdot \left(-x^2-4x+5\right)\\ =-3x^4+7x^2-12x^3+28x+15x^2-35\\ =-3x^4-12x^3+22x^2+28x-35\end{array}$

Simplifying further,

  $\left(fg\right)\left(x\right)=-3x^4-12x^3+22x^2+28x-35$

Hence, theexpression of the function $\left(fg\right)\left(x\right)$ is $-3x^4-12x^3+22x^2+28x-35$ .

(d)

To determine

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

Answer to Problem 19CT

The expression of the function $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{\left(3x^2-7\right)}{-\left(x+5\right)\left(x-1\right)}$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=3x^2-7$

And,

  $g\left(x\right)=-x^2-4x+5$

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,

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{\left(3x^2-7\right)}{\left(-x^2-4x+5\right)}\\ =\frac{\left(3x^2-7\right)}{-\left(x^2+5x-x-5\right)}\\ =\frac{\left(3x^2-7\right)}{-\left(x\left(x+5\right)-1\left(x+5\right)\right)}\end{array}$

Simplifying further,

  $\left(\frac{f}{g}\right)\left(x\right)=\frac{\left(3x^2-7\right)}{-\left(x+5\right)\left(x-1\right)}$

Hence, the expression of the function $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{\left(3x^2-7\right)}{-\left(x+5\right)\left(x-1\right)}$ .

(e)

To determine

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

Answer to Problem 19CT

The expression of the function $\left(f\circ g\right)\left(x\right)$ is $9x^4-54x^2+75$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=3x^2-7$

And,

  $g\left(x\right)=-x^2-4x+5$

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,

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ =f\left(-x^2-4x+5\right)\\ ={\left(3x^2-7\right)}^2-4\left(3x^2-7\right)+5\\ =9x^4+49-42x^2-12x^2+21+5\end{array}$

Simplifying further,

  $\left(f\circ g\right)\left(x\right)=9x^4-54x^2+75$

Hence, the expression of the function $\left(f\circ g\right)\left(x\right)$ is $9x^4-54x^2+75$ .

(f)

To determine

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

Answer to Problem 19CT

The expression of the function $\left(g\circ f\right)\left(x\right)$ is $3x^4+24x^3+18x^2-120x+68$ .

Explanation of Solution

Given information:

Consider the given function,

  $f\left(x\right)=3x^2-7$

And,

  $g\left(x\right)=-x^2-4x+5$

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,

  $\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\\ =g\left(3x^2-7\right)\\ =3{\left(-x^2-4x+5\right)}^2-7\\ =3x^4+24x^3+18x^2-120x+75-7\end{array}$

Simplifying further,

  $\left(g\circ f\right)\left(x\right)=3x^4+24x^3+18x^2-120x+68$

Hence, the expression of the function $\left(g\circ f\right)\left(x\right)$ is $3x^4+24x^3+18x^2-120x+68$ .