Chapter 2.2, Problem 35AYU

(a)

To determine

To illustrated: the graph of $\left(f+g\right)\left(2\right)$ .

Answer to Problem 35AYU

  $\left(f+g\right)\left(2\right)=3$

Explanation of Solution

Given information:

The graph of two functions $f$ and $g$

Given graph

  

Calculation:

The sum of function can be defined as $\left(f+g\right)\left(2\right)=f\left(2\right)+g\left(2\right)$

The point $\left(2,2\right)$ on the graph corresponds to the value of the function $f\left(x\right)$ at $x=2$ .

So, $f\left(2\right)=2$ .

The point $\left(2,1\right)$ on the graph corresponds to the value of the function $g\left(x\right)$ at $x=2$ .

Thus, $g\left(2\right)=1$ .

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

Therefore, $\left(f+g\right)\left(2\right)=3$

(b)

To determine

To illustrated: the graph of $\left(f+g\right)\left(4\right)$ .

Answer to Problem 35AYU

  $\left(f+g\right)\left(4\right)=-2$

Explanation of Solution

Given information:

The graph of two functions $f$ and $g$

Given graph

  

Calculation:

The sum of function can be defined as $\left(f+g\right)\left(4\right)=f\left(4\right)+g\left(4\right)$

The point $\left(4,1\right)$ on the graph corresponds to the value of the function $f\left(x\right)$ at $x=4$ .

So, $f\left(4\right)=1$ .

The point $\left(4,-3\right)$ on the graph corresponds to the value of the function $g\left(x\right)$ at $x=4$ .

Thus, $g\left(4\right)=-3$ .

  $\begin{array}{c}f\left(2\right)+g\left(2\right)=1+\left(-3\right)\\ =-2\end{array}$

Therefore, $\left(f+g\right)\left(4\right)=-2$

(c)

To determine

To illustrated: the graph of $\left(f-g\right)\left(6\right)$ .

Answer to Problem 35AYU

  $\left(f-g\right)\left(6\right)=-1$

Explanation of Solution

Given information:

The graph of two functions $f$ and $g$

Given graph

  

Calculation:

The difference function can be defined as $\left(f-g\right)\left(6\right)=f\left(6\right)-g\left(6\right)$

The point $\left(6,0\right)$ on the graph corresponds to the value of the function $f\left(x\right)$ at $x=6$ .

So, $f\left(6\right)=0$ .

The point $\left(6,1\right)$ on the graph corresponds to the value of the function $g\left(x\right)$ at $x=6$ .

Thus, $g\left(6\right)=1$ .

  $\begin{array}{c}f\left(6\right)-g\left(6\right)=0-1\\ =-1\end{array}$

Therefore, $\left(f-g\right)\left(6\right)=-1$

(d)

To determine

To illustrated: the graph of $\left(g-f\right)\left(6\right)$ .

Answer to Problem 35AYU

  $\left(g-f\right)\left(6\right)=1$

Explanation of Solution

Given information:

The graph of two functions $f$ and $g$

Given graph

  

Calculation:

The difference function can be defined as $\left(g-f\right)\left(6\right)=g\left(6\right)-f\left(6\right)$

The point $\left(6,1\right)$ on the graph corresponds to the value of the function $g\left(x\right)$ at $x=6$ .

So, $g\left(6\right)=1$ .

The point $\left(6,0\right)$ on the graph corresponds to the value of the function $f\left(x\right)$ at $x=6$ .

Thus, $f\left(6\right)=0$ .

  $\begin{array}{c}g\left(6\right)-f\left(6\right)=1-0\\ =1\end{array}$

Therefore, $\left(g-f\right)\left(6\right)=1$

(e)

To determine

To illustrated: the graph of $\left(f\cdot g\right)\left(2\right)$ .

Answer to Problem 35AYU

  $\left(f\cdot g\right)\left(2\right)=2$

Explanation of Solution

Given information:

The graph of two functions $f$ and $g$

Given graph

  

Calculation:

The product function can be defined as $\left(f\cdot g\right)\left(2\right)=f\left(2\right)\cdot g\left(2\right)$

The point $\left(2,2\right)$ on the graph corresponds to the value of the function $f\left(x\right)$ at $x=2$ .

So, $f\left(2\right)=2$ .

The point $\left(2,1\right)$ on the graph corresponds to the value of the function $g\left(x\right)$ at $x=2$ .

Thus, $g\left(2\right)=1$ .

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

Therefore, $\left(f\cdot g\right)\left(2\right)=2$

(f)

To determine

To illustrated: the graph of $\left(\frac{f}{g}\right)\left(4\right)$ .

Answer to Problem 35AYU

  $\left(\frac{f}{g}\right)\left(4\right)=-\frac{1}{3}$

Explanation of Solution

Given information:

The graph of two functions $f$ and $g$

Given graph

  

Calculation:

The quotient of function can be defined as $\left(\frac{f}{g}\right)\left(4\right)=\frac{f\left(4\right)}{g\left(4\right)}$

The point $\left(4,1\right)$ on the graph corresponds to the value of the function $f\left(x\right)$ at $x=4$ .

So, $f\left(4\right)=1$ .

The point $\left(4,-3\right)$ on the graph corresponds to the value of the function $g\left(x\right)$ at $x=4$ .

Thus, $g\left(4\right)=-3$ .

  $\begin{array}{c}\frac{f\left(4\right)}{g\left(4\right)}=\frac{1}{-3}\\ =-\frac{1}{3}\end{array}$

Therefore, $\left(\frac{f}{g}\right)\left(4\right)=-\frac{1}{3}$