Chapter 1, Problem 8RCC

(a)

To determine

To sketch: The rough graph of the function $y=\sin x$.

Explanation of Solution

The rough graph of $y=\sin x$ is roughly drawn and shown below in Figure 1.

From Figure 1, it is observed that the sine function is defined on $-1\le \sin x\le 1$ and is a periodic function with period $2\pi ,\sin \left(x+2\pi \right)=\sin \left(x\right)$.

(b)

To determine

To sketch: The rough graph of the function $y=\tan x$.

Explanation of Solution

The rough graph of $y=\tan x$ is roughly drawn and shown below in Figure 2.

From Figure 2, it is observed that the tan function is undefined when cos x = 0 and is a periodic with period $\pi ,\tan \left(x+\pi \right)=\tan \left(x\right)$.

(c)

To determine

To sketch: The rough graph of the function $y=e^x$.

Explanation of Solution

The rough graph of $y=e^x$ is roughly drawn and shown below in Figure 3.

From Figure 3, it is observed that the exponential function is an increasing function with the domain $\left(-\infty ,\infty \right)$ and the range $\left(0,\infty \right)$.

(d)

To determine

To sketch: The rough graph of the function $y=\ln x$.

Explanation of Solution

The rough graph of $y=\ln x$ is roughly drawn and shown below in Figure 4.

From Figure 4, it is observed that the logarithmic function is an increasing function with the domain $\left(0,\infty \right)$ and the range $\left(-\infty ,\infty \right)$. Also, notice that the logarithmic function is a reciprocal of an exponential function.

(e)

To determine

To sketch: The rough graph of the function $y=\frac{1}{x}$.

Explanation of Solution

The rough graph of $y=\frac{1}{x}$ is roughly drawn and shown below in Figure 5.

From Figure 5, it is observed that the reciprocal function is an hyperbola with x and y axes as its asymptotes.

(f)

To determine

To sketch: The rough graph of the function $y=\left|x\right|$.

Explanation of Solution

The rough graph of $y=\left|x\right|$ is roughly drawn and shown below in Figure 6.

From Figure 6, it is observed that the range of an absolute function is positive y-axis.

(g)

To determine

To sketch: The rough graph of the function $y=\sqrt{x}$.

Explanation of Solution

The rough graph of $y=\sqrt{x}$ is roughly drawn and shown below in Figure 7.

From Figure 7, it is observed that the range of the power function with the power $\frac{1}{2}$.

(h)

To determine

To sketch: The rough graph of the function $y={\tan }^{-1}x$.

Explanation of Solution

The rough graph of $y={\tan }^{-1}x$ is roughly drawn and shown below in Figure 8.

From Figure 8, it is observed that the domain of an inverse tan function is $\left(-\infty ,\infty \right)$ and the range is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Also, notice that the tan function and the inverse of trigonometric function are inverse function of each other. So, they reflect about the y = x.