Chapter 1.2, Problem 1E

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

1. (a) $f\left(x\right)={\log }_{2}x$

(b) $g(x)=\sqrt[4]{x}$

(c) $h(x)=\frac{2x^3}{1-x^2}$

(d) $u(t)=1-1.1t+2.54t^2$

(e) $v(t)=5^t$

(f) $w(\theta )=\sin \theta co{s}^2\theta$

To determine

To classify: The function $f(x)={\log }_{2}x$ as one of the type of functions.

Answer to Problem 1E

The function $f(x)={\log }_{2}x$ is a logarithmic function.

Explanation of Solution

Reason:

The function is of the form $f(x)={\log }_{b}x$, where b is a positive constant is said to be a logarithmic function. It is a reciprocal of an exponential function.

To determine

To classify: The function $g(x)=\sqrt[4]{x}$ as one of the type of functions.

Answer to Problem 1E

The function $g(x)=\sqrt[4]{x}$ is a root function.

Explanation of Solution

Reason:

The function is of the form $f\left(x\right)=\sqrt[n]{x}$, is said to be a root function. Note that the expression inside the root is always positive.

To determine

To classify: The function $h(x)=\frac{2x^3}{1-x^2}$ as one of the type of functions.

Answer to Problem 1E

The function $h(x)=\frac{2x^3}{1-x^2}$ is a rational function.

Explanation of Solution

Reason:

The function is of the form $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, where $p\left(x\right)$ and $q\left(x\right)$ are the polynomials and $q\left(x\right)\ne 0$ is said to be a rational function. Here, $p\left(x\right)=2x^3$ and $q\left(x\right)=1-x^2$. Thus, the function $h(x)$ is a rational function.

To determine

To classify: The function $u(t)=1-1.1t+2.54t^2$ as one of the type of functions.

Answer to Problem 1E

The function $u(t)=1-1.1t+2.54t^2$ is a polynomial function with degree 2.

Explanation of Solution

Reason:

Rewrite the given function as $u(t)=2.54t^2-1.1t+1$.

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 2. Therefore, the function is a quadratic function.

To determine

To classify: The function $v(t)=5^t$ as one of the type of functions.

Answer to Problem 1E

The function $v(t)=5^t$ 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, $b=5$ and $x=t$. Thus, the given function is said to be an exponential function.

To determine

To classify: The function $w(\theta )=\sin \theta {\cos }^2\theta$ as one of the type of functions.

Answer to Problem 1E

The function $w(\theta )=\sin \theta {\cos }^2\theta$ is a trigonometric function.

Explanation of Solution

Reason:

The function involves with the trigonometric functions such as $\sin \theta \text{ and }\cos \theta$. Thus, the given function is said to bea trigonometric function.