Question

question 33 of 42
which of the following is a rational function?

a. $f(x) = \frac{x^2 - 9}{7x}$

b. $f(x) = 2x + 3$

c. $f(x) = -x^3 + 6x^2 - 3$

d. $f(x) = \sqrt{x - 5}$

Explanation:

Step1: Recall rational function definition

A rational function is a function of the form \( F(x)=\frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x)
eq0 \).

Step2: Analyze Option A

For \( F(x)=\frac{x^{2}-9}{7x} \), \( P(x)=x^{2}-9 \) (a polynomial) and \( Q(x)=7x \) (a polynomial, \( 7x
eq0 \) when \( x
eq0 \)). So it fits the rational function form.

Step3: Analyze Option B

\( F(x)=2x + 3 \) is a linear polynomial function, not a rational function (denominator is 1, but it's a polynomial, not in the form of a ratio of two non - constant polynomials in the typical rational function context here as it can be written as \( \frac{2x + 3}{1} \), but usually rational functions have non - constant denominator polynomial. However, strictly by definition, it is a rational function, but let's check other options). Wait, no, actually, a polynomial is a special case of a rational function (with denominator 1), but let's check other options.

Step4: Analyze Option C

\( F(x)=-x^{3}+6x^{2}-3 \) is a polynomial function, not a rational function in the sense of a ratio of two polynomials (unless we consider it as \( \frac{-x^{3}+6x^{2}-3}{1} \), but the other options have non - trivial denominators. However, let's check Option D.

Step5: Analyze Option D

\( F(x)=\sqrt{x - 5} \) is a square - root function, not a rational function as it involves a radical, not a ratio of polynomials.

Now, between Option A and Option B: A rational function is usually defined as a ratio of two polynomials where the denominator is not a constant polynomial (or at least, the non - trivial case). \( F(x)=\frac{x^{2}-9}{7x} \) has a non - constant polynomial in the denominator, while \( F(x)=2x + 3=\frac{2x + 3}{1} \) has a constant polynomial in the denominator. So the more typical rational function (with non - constant denominator) is Option A.

Answer:

A. \( F(x)=\frac{x^{2}-9}{7x} \)