Chapter 2.6, Problem 1E

From the graphs of f and g in the figure, we find

(f + g)(2) = _____

(f − g)(2) = _____

(fg)(2) = _____

$\left(\frac{f}{g}\right)(2)=$_____

To determine

To evaluate: The values of functions $\left(f+g\right)\left(2\right)$, $\left(f-g\right)\left(2\right)$, $\left(f\cdot g\right)\left(2\right)$ and $\left(\frac{f}{g}\right)\left(2\right)$ from the given graph.

Answer to Problem 1E

The values of $\left(f+g\right)\left(2\right)$ is 8, $\left(f-g\right)\left(2\right)$ is $\underset{\_}{-2}$ $\left(f\cdot g\right)\left(2\right)$ is 15 and $\left(\frac{f}{g}\right)\left(2\right)$ is $\frac{3}{5}$.

Explanation of Solution

Given:

The graphs of f and g is given below,

Figure (1)

Graphs of function $f\left(x\right)$ and $g\left(x\right)$ are shown in Figure (1).

Calculation:

From the Figure (1), in the graph of $y=f\left(x\right)$, the value of 2 at x-axis is at 3 of y-axis, so the value of $f\left(2\right)$ is 3.

From the Figure (1), in the graph of $y=g\left(x\right)$, the value of 2 at x-axis is at 5 of y-axis, so the value of $g\left(2\right)$ is 5.

Calculate the sum of the both the functions by below formula,

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

Substitute 2 for x in above formula,

$\left(f+g\right)\left(2\right)=f\left(2\right)+g\left(2\right)$.

Substitute 3 for $f\left(2\right)$ and 5 for $g\left(2\right)$ in above formula,

$\begin{array}{c}\left(f+g\right)\left(2\right)=3+5\\ =8\end{array}$

Thus, the values of $\left(f+g\right)\left(2\right)$ is 8.

Now, from the Figure (1), in the graph of $y=f\left(x\right)$, the value of 2 at x-axis is at 3 of y-axis, so the value of $f\left(2\right)$ is 3.

From the Figure (1), in the graph of $y=g\left(x\right)$, the value of 2 at x-axis is at 5 of y-axis, so the value of $g\left(2\right)$ is 5.

Calculate the difference of the both the functions by below formula,

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

Substitute 2 for x in above formula,

$\left(f-g\right)\left(2\right)=f\left(2\right)-g\left(2\right)$.

Substitute 3 for $f\left(2\right)$ and 5 for $g\left(2\right)$ in above formula,

$\begin{array}{c}\left(f-g\right)\left(2\right)=3-5\\ =-2\end{array}$

Thus, the value of $\left(f-g\right)\left(2\right)$ is $\underset{\_}{-2}$

Similarly, From the Figure (1), in the graph of $y=f\left(x\right)$, the value of 2 at x-axis is at 3 of y-axis, so the value of $f\left(2\right)$ is 3.

From the Figure (1), in the graph of $y=g\left(x\right)$, the value of 2 at x-axis is at 5 of y-axis, so the value of $g\left(2\right)$ is 5.

Calculate the product of two functions by below formula,

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

Substitute 2 for x in above formula,

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

Substitute 3 for $f\left(2\right)$ and 5 for $g\left(2\right)$ in above formula,

$\begin{array}{c}\left(f\cdot g\right)\left(2\right)=3\cdot 5\\ =15\end{array}$

Thus, the value of $\left(f\cdot g\right)\left(2\right)$ is 15.

From the Figure (1), in the graph of $y=f\left(x\right)$, the value of 2 at x-axis is at 3 of y-axis, so the value of $f\left(2\right)$ is 3.

From the Figure (1), in the graph of $y=g\left(x\right)$, the value of 2 at x-axis is at 5 of y-axis, so the value of $g\left(2\right)$ is 5.

Calculate the product of two functions by below formula,

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

Substitute 2 for x in above formula,

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

Substitute 3 for $f\left(2\right)$ and 5 for $g\left(2\right)$ in above formula,

$\left(\frac{f}{g}\right)\left(2\right)=\frac{3}{5}$

Thus, the value of $\left(\frac{f}{g}\right)\left(2\right)$ is $\frac{3}{5}$.