Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Feb 02, 2024

Identify the polynomial function from the given options: $f(x)=x^{2}+4 x-\frac{7}{x}$ , $f(x)=3 x^{3}-2 x^{2}+\sqrt{x}$ , $f(x)=-5 x^{-4}-3^{2}$ , $f(x)=2 x^{2}-\sqrt{7}$ , $f(x)=\frac{x+3}{2 x-6}$ .

STEP 1

Assumptions
1. A polynomial function is a function that can be expressed in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$ , where $n$ is a non-negative integer and the coefficients $a_n, a_{n-1}, \ldots, a_1, a_0$ are real numbers.
2. The powers of $x$ in a polynomial function must be non-negative integers.
3. Each option given must be checked to see if it satisfies the definition of a polynomial function.

STEP 2

Check option (a) $f(x)=x^{2}+4x-\frac{7}{x}$ .
Notice that $\frac{7}{x}$ is equivalent to $7x^{-1}$ . Since the exponent $-1$ is not a non-negative integer, this term does not satisfy the definition of a polynomial.
$f(x)=x^{2}+4x-\frac{7}{x} \text{ is not a polynomial function.}$

STEP 3

Check option (b) $f(x)=3x^{3}-2x^{2}+\sqrt{x}$ .
Notice that $\sqrt{x}$ is equivalent to $x^{1/2}$ . Since the exponent $1/2$ is not an integer, this term does not satisfy the definition of a polynomial.
$f(x)=3x^{3}-2x^{2}+\sqrt{x} \text{ is not a polynomial function.}$

STEP 4

Check option (c) $f(x)=-5x^{-4}-3^{2}$ .
Notice that $-5x^{-4}$ has an exponent of $-4$ , which is not a non-negative integer. Additionally, $3^2$ is a constant term, but the presence of $-5x^{-4}$ disqualifies the function from being a polynomial.
$f(x)=-5x^{-4}-3^{2} \text{ is not a polynomial function.}$

STEP 5

Check option (d) $f(x)=2x^{2}-\sqrt{7}$ .
Here, $2x^{2}$ has an exponent of $2$ , which is a non-negative integer, and $-\sqrt{7}$ is a constant term. Since all terms satisfy the definition of a polynomial, this function is a polynomial.
$f(x)=2x^{2}-\sqrt{7} \text{ is a polynomial function.}$

STEP 6

Since we have found a polynomial function in option (d), we do not need to check option (e). However, for completeness, let's analyze it.
Check option (e) $f(x)=\frac{x+3}{2x-6}$ .
This function is a rational function, not a polynomial, because it involves division by a term that includes the variable $x$ .
$f(x)=\frac{x+3}{2x-6} \text{ is not a polynomial function.}$

STEP 7

The correct answer is option (d) $f(x)=2x^{2}-\sqrt{7}$ , which is a polynomial function.
The solution to the problem is option (d).

Was this helpful?

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Contact Influencer program Policy Terms