Chapter 1.2, Problem 2E

(a)

To determine

To classify: The function $y={\pi }^x$ as one of the type of functions.

Answer to Problem 2E

The function $y={\pi }^x$ is an exponential function.

Explanation of Solution

Reason:

An exponential function is of the form $f\left(x\right)=b^x$, where b is a positive constant and x is an exponent. Here, $b=\pi$. Thus, the given function is said to be an exponential function.

(b)

To determine

To classify: The function $y=x^{\pi }$ as one of the type of functions.

Answer to Problem 2E

The function $y=x^{\pi }$ is a power function.

Explanation of Solution

Reason:

The power function $y=x^{\pi }$ is of the form $f\left(x\right)=x^n$, where n is a positive integer. Here, the variable x is called as base. Thus, the given function is said to be a power function.

(c)

To determine

To classify: The function $y=x^2\left(2-x^3\right)$ as one of the type of functions.

Answer to Problem 2E

The function $y=x^2\left(2-x^3\right)$ is a polynomial function.

Explanation of Solution

Reason:

Rewrite the given function as $y=2x^2-x^5$.

The polynomial function is of the form $p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$, where $n$ is a positive integer and $a_0,a_1,\mathrm{...},a_n$ are constants. Thus, the given function is a polynomial function with degree 5.

(d)

To determine

To classify: The function $y=\tan t-\cos t$ as one of the type of functions.

Answer to Problem 2E

The function $y=\tan t-\cos t$ is a trigonometric function.

Explanation of Solution

Reason:

The function involves with the trigonometric functions such as $\cos t\text{ and }\tan t$. Thus, the given function is said to be a trigonometric function.

(e)

To determine

To classify: The function $y=\frac{s}{1+s}$ as one of the type of functions.

Answer to Problem 2E

The function $y=\frac{s}{1+s}$ is a rational function.

Explanation of Solution

Reason:

The function is of the form $f\left(s\right)=\frac{p\left(s\right)}{q\left(s\right)}$, where $p\left(s\right)$ and $q\left(s\right)$ are the polynomials and $q\left(s\right)\ne 0$ is said to be a rational function. Here, $p\left(s\right)=s$ and $q\left(s\right)=1+s$. Thus, the given function is a rational function.

(f)

To determine

To classify: The function $y=\frac{\sqrt{x^3-1}}{1+\sqrt[3]{x}}$ as one of the type of functions.

Answer to Problem 2E

The function $y=\frac{\sqrt{x^3-1}}{1+\sqrt[3]{x}}$ is an algebraic function.

Explanation of Solution

Reason:

The function $y=\frac{\sqrt{x^3-1}}{1+\sqrt[3]{x}}$ involves with polynomial and root functions. Thus, the given function is an algebraic function.