Q 35. - Precalculus Enhanced with Graphing Utilities

Question

The graph of two functions, f and g, is illustrated. 

Use the graph to answer parts (a) – (f).

$$\begin{gathered} \left(a\right) (f + g)\left(2\right) \left(b\right) (f + g)\left(4\right) \\ \left(c\right) (f - g)\left(6\right) \left(d\right) (g-f)\left(6\right) \\ \left(e\right) (f · g)\left(2\right) \left(f\right) \left(\frac{f}{g}\right)\left(4\right) \end{gathered}$$

Step-by-Step Solution

Answer

From the graph,

$$\begin{gathered} \mathrm{Part} \left(\mathrm{a}\right) (f + g)\left(2\right)=3 \\ \mathrm{Part} \left(\mathrm{b}\right) (f + g)\left(4\right)=-2 \\ \mathrm{Par}t \left(\mathrm{c}\right) (f - g)\left(6\right)=-1 \\ \mathrm{Part} \left(\mathrm{d}\right) (g-f)\left(6\right)=1 \\ \mathrm{Part} \left(\mathrm{e}\right) (f · g)\left(2\right)=2 \\ \mathrm{Part} \left(\mathrm{f}\right) \left(\frac{f}{g}\right)\left(4\right)=-\frac{1}{3} \end{gathered}$$

Part (a) Step 1. Given information.

We have given graphs are f(x) and g(x) is,

Part (a) Step 2. Concept used.

Functions operation:

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

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

Multiplication: $(f·g)\left(x\right)=f\left(x\right)·g\left(x\right)$

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

Part (a) Step 3. Explanation.

From the graph of f(x) and g(x) we have,

$f\left(2\right)=2 and g\left(2\right)=1$

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

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

Part (a) Step 4. Conclusion.

Hence, value for $(f+g)\left(2\right)$ is 3.

Part (b) Step 1. Explanation.

From the graphs of f(x) and g(x) we have,

$f\left(4\right)=1 and g\left(4\right) = -3$.

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

$$\begin{gathered} (f+g)\left(4\right)=f\left(4\right)+g\left(4\right) \\ =1+(-3) \\ =-2 \end{gathered}$$

Part (b) Step 2. Conclusion.

Hence, value for $(f+g)\left(4\right)$ is -2.

Part (c) Step 1. Explanation.

From the graphs of f(x) and g(x) we have,

$f\left(6\right)=0 and g\left(6\right) = 1$

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

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

Part (c) Step 2. Conclusion.

Hence, the value for $(f-g)\left(6\right)$ is -1.

Part (d) Step 1. Explanation.

From the graphs of f(x) and g(x) we have,

$f\left(6\right)=0 and g\left(6\right)=1$.

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

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

Part (d) Step 2. Conclusion.

Hence, value for $(g-f)\left(6\right)$ is 1.

Part (e) Step 1. Explanation.

From graphs of the f(x) and g(x) we have,

$f\left(2\right)=2 and g\left(2\right)=1$.

Using, $(f·g)\left(x\right)=f\left(x\right)·g\left(x\right)$,

$$\begin{gathered} (f·g)\left(2\right)=f\left(2\right)·g\left(2\right) \\ =2·1 \\ =2 \end{gathered}$$

Part (e) Step 2. Conclusion.

Hence, value for $(f·g)\left(2\right)$ is 2.

Part (f) Step 1. Explanation.

From the given graphs we have,

$f\left(4\right)=1 and g\left(4\right) =-3$.

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

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

Part (f) Step 2. Conclusion.

Hence, value for $\left(\frac{f}{g}\right)\left(4\right)$ is $-\frac{1}{3}$.