Chapter 4.1, Problem 2E

To determine

To choose: the function which is different from other three functions.

Answer to Problem 2E

The function which is different from other three functions is $h\left(x\right)=-3x^4+5x^{-1}-3x^2$ .

Explanation of Solution

Given information:

Four functions are given as

  $f\left(x\right)=7x^5+3x^2-2x$ , $g\left(x\right)=-3x^3-2x^8+\frac{3}{4}$ , $h\left(x\right)=-3x^4+5x^{-1}-3x^2$ , and

  $k\left(x\right)=\sqrt{3}x+8x^4+2x+1$

Calculation:

A polynomial function is of the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$

Terms of a polynomial function should be arranged in descending order according to its degree to express it in a standard form and degree of each term should be a positive integer or whole number. The coefficients should be real numbers.

Now, for the functions $f\left(x\right)=7x^5+3x^2-2x$ , $g\left(x\right)=-3x^3-2x^8+\frac{3}{4}$ , and $k\left(x\right)=\sqrt{3}x+8x^4+2x+1$ degree of each term is a whole number.

So, the three functions are polynomial functions.

But, for the function $h\left(x\right)=-3x^4+5x^{-1}-3x^2$ , degree of one term $5x^{-1}$ is not an whole number.

So, it is not a polynomial function.

Hence, the function which is different from other three functions is $h\left(x\right)=-3x^4+5x^{-1}-3x^2$ .