Answer :
To determine which of the given functions is a rational function, let's review the definition of a rational function:
A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it has the form [tex]\( \frac{P(x)}{Q(x)} \)[/tex], where [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] are polynomials, and [tex]\( Q(x) \neq 0 \)[/tex].
Now, let's examine each of the given options:
### Option A: [tex]\( F(x) = -7x + 15 \)[/tex]
- This function is a linear function, represented as a polynomial of degree 1.
- It can be written as [tex]\( \frac{-7x + 15}{1} \)[/tex], which is a valid ratio but not in the form of a ratio of polynomials where the denominator is higher than just a constant.
- Therefore, [tex]\( F(x) = -7x + 15 \)[/tex] is a polynomial, not a rational function in its general sense.
### Option B: [tex]\( F(x) = \sqrt{2x - 1} + 9 \)[/tex]
- This function involves a square root, specifically [tex]\( \sqrt{2x - 1} \)[/tex].
- Since rational functions are defined by ratios of polynomials and polynomials do not include square roots as basic components, this function is not a polynomial.
- Thus, [tex]\( F(x) = \sqrt{2x - 1} + 9 \)[/tex] is not a rational function.
### Option C: [tex]\( F(x) = \frac{x^2 + 5x - 6}{3x} \)[/tex]
- This function is explicitly written as the ratio of two polynomials.
- The numerator [tex]\( P(x) = x^2 + 5x - 6 \)[/tex] is a polynomial of degree 2.
- The denominator [tex]\( Q(x) = 3x \)[/tex] is a polynomial of degree 1.
- Since both the numerator and the denominator are polynomials, this meets the criteria for being a rational function.
- Therefore, [tex]\( F(x) = \frac{x^2 + 5x - 6}{3x} \)[/tex] is a rational function.
### Option D: [tex]\( F(x) = x^4 - x^2 + 8 \)[/tex]
- This function is a polynomial consisting of several terms with different degrees.
- It can be written in the form [tex]\( P(x) = x^4 - x^2 + 8 \)[/tex] with no denominator other than 1, so it is not expressed as a ratio.
- Therefore, [tex]\( F(x) = x^4 - x^2 + 8 \)[/tex] is a polynomial, not a rational function.
### Conclusion:
Among the provided options, the only function that meets the definition of a rational function is:
Option C: [tex]\( F(x) = \frac{x^2 + 5x - 6}{3x} \)[/tex].
A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it has the form [tex]\( \frac{P(x)}{Q(x)} \)[/tex], where [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] are polynomials, and [tex]\( Q(x) \neq 0 \)[/tex].
Now, let's examine each of the given options:
### Option A: [tex]\( F(x) = -7x + 15 \)[/tex]
- This function is a linear function, represented as a polynomial of degree 1.
- It can be written as [tex]\( \frac{-7x + 15}{1} \)[/tex], which is a valid ratio but not in the form of a ratio of polynomials where the denominator is higher than just a constant.
- Therefore, [tex]\( F(x) = -7x + 15 \)[/tex] is a polynomial, not a rational function in its general sense.
### Option B: [tex]\( F(x) = \sqrt{2x - 1} + 9 \)[/tex]
- This function involves a square root, specifically [tex]\( \sqrt{2x - 1} \)[/tex].
- Since rational functions are defined by ratios of polynomials and polynomials do not include square roots as basic components, this function is not a polynomial.
- Thus, [tex]\( F(x) = \sqrt{2x - 1} + 9 \)[/tex] is not a rational function.
### Option C: [tex]\( F(x) = \frac{x^2 + 5x - 6}{3x} \)[/tex]
- This function is explicitly written as the ratio of two polynomials.
- The numerator [tex]\( P(x) = x^2 + 5x - 6 \)[/tex] is a polynomial of degree 2.
- The denominator [tex]\( Q(x) = 3x \)[/tex] is a polynomial of degree 1.
- Since both the numerator and the denominator are polynomials, this meets the criteria for being a rational function.
- Therefore, [tex]\( F(x) = \frac{x^2 + 5x - 6}{3x} \)[/tex] is a rational function.
### Option D: [tex]\( F(x) = x^4 - x^2 + 8 \)[/tex]
- This function is a polynomial consisting of several terms with different degrees.
- It can be written in the form [tex]\( P(x) = x^4 - x^2 + 8 \)[/tex] with no denominator other than 1, so it is not expressed as a ratio.
- Therefore, [tex]\( F(x) = x^4 - x^2 + 8 \)[/tex] is a polynomial, not a rational function.
### Conclusion:
Among the provided options, the only function that meets the definition of a rational function is:
Option C: [tex]\( F(x) = \frac{x^2 + 5x - 6}{3x} \)[/tex].
