To determine which of the given functions is a rational function, we need to understand that a rational function is defined as the quotient of two polynomials. Let's analyze each function step by step:
Option A: [tex]\( F(x) = \frac{x^2 - 9}{7x} \)[/tex]
This function is a ratio of two polynomials:
- The numerator [tex]\( x^2 - 9 \)[/tex] is a polynomial.
- The denominator [tex]\( 7x \)[/tex] is a polynomial.
Since both parts are polynomials and it is a quotient of these polynomials, this function is indeed a rational function.
Option B: [tex]\( F(x) = 2x + 3 \)[/tex]
This function represents a linear polynomial, which is a specific type of polynomial but not a rational function because it is not expressed as the ratio of two polynomials. It is a single polynomial term.
Option C: [tex]\( F(x) = \sqrt{x - 5} \)[/tex]
This function contains a square root. Functions that involve square roots (or any roots) of [tex]\( x \)[/tex] are not considered rational functions because they cannot be expressed as the quotient of two polynomials.
Option D: [tex]\( F(x) = -x^3 + 6x^2 - 3 \)[/tex]
This function is a polynomial function of degree 3. It’s a single polynomial expression and not the quotient of two polynomials, so it is not considered a rational function.
From our analysis, the only function that meets the criteria of being a quotient of two polynomials is Option A.
Therefore, the correct answer is:
A. [tex]\( F(x) = \frac{x^2-9}{7x} \)[/tex]
