Chapter 1, Problem 6RCC

(a)

To determine

To give: An example of linear function.

Answer to Problem 6RCC

Solution:

The function $y=x$ is a linear function.

Explanation of Solution

Reason:

A linear function is of the form $f\left(x\right)=mx+c,a\ne 0$, where m is the slop and c is the y-intercept. Thus, the function $y=x$ is said to be linear. The degree of the linear function is always 1.

(b)

To determine

To give: An example of power function.

Answer to Problem 6RCC

Solution:

The function $y=x^6$ is a power function.

Explanation of Solution

Reason:

The power function is of the form $f\left(x\right)=x^n$, where n is a positive integer,  the variable x is called as base. Thus, the function $y=x^6$ is said to be a power function.

(c)

To determine

To give: An example of exponential function.

Answer to Problem 6RCC

Solution:

The function $y=2^x$ 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, $b=2$. Thus, the function $y=2^x$ is said to be an exponential function.

(d)

To determine

To give: An example of quadratic function.

Answer to Problem 6RCC

Solution:

The function $y=x^2\left(2-\frac{3}{x}\right)$ is a quadratic function.

Explanation of Solution

Reason:

Rewrite the given function as $y=2x^2-3x$.

A quadratic function is of the form $f\left(x\right)=a{x}^2+bx+c,a\ne 0$. Thus, the function $y=x^2\left(2-\frac{3}{x}\right)$ is a quadratic function. A function of degree 2 is a quadratic function.

(e)

To determine

To give: An example of polynomial of degree 5.

Answer to Problem 6RCC

Solution:

The function $v(t)=9+2.54t^2-3t^3+\frac{1}{2}{t}^5$ is a polynomial function of degree 5.

Explanation of Solution

Reason:

Rewrite the given function as $v(t)=\frac{1}{2}{t}^5-3t^3+2.54t^2+9$.

The 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 is said to be polynomial function. Thus, the function $v(t)=9+2.54t^2-3t^3+\frac{1}{2}{t}^5$ is said to be a polynomial function. Observe that the maximum degree of $v\left(t\right)$ is 5.

(f)

To determine

To give: An example of rational function.

Answer to Problem 6RCC

Solution:

The function $y=\frac{x}{3+x}$ is a rational function.

Explanation of Solution

Reason:

A 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. Thus, the function $y=\frac{x}{3+x}$ is said to be rational function.