To determine which of the given functions is a polynomial, we need to understand the definition of a polynomial function. A polynomial function is expressed as a finite sum of terms, each term being a product of a constant and a non-negative integer power of [tex]\( x \)[/tex].
Let's analyze each function:
1. [tex]\( f(x) = 5x^3 + 2x^2 - 6x + 7 \)[/tex]:
- This function is composed of terms like [tex]\( 5x^3 \)[/tex], [tex]\( 2x^2 \)[/tex], [tex]\( -6x \)[/tex], and [tex]\( 7 \)[/tex].
- Each term consists of a constant multiplied by [tex]\( x \)[/tex] raised to a whole number (non-negative integer) power.
- Hence, [tex]\( f(x) \)[/tex] fits the definition of a polynomial function.
2. [tex]\( g(x) = \frac{x^2 + 2x - 3}{x - 4} \)[/tex]:
- This function involves a division of polynomials.
- The presence of the denominator [tex]\( (x - 4) \)[/tex] means that the function can have a vertical asymptote where [tex]\( x = 4 \)[/tex], indicating a discontinuity.
- Since polynomials cannot have such discontinuities and must be defined for all real numbers, [tex]\( g(x) \)[/tex] is a rational function, not a polynomial function.
3. [tex]\( h(x) = 750(1.04)^x \)[/tex]:
- This function includes an exponential term [tex]\( (1.04)^x \)[/tex].
- Polynomial functions cannot have terms where the variable [tex]\( x \)[/tex] is in the exponent.
- Thus, [tex]\( h(x) \)[/tex] is not a polynomial function.
4. [tex]\( p(x) = \log_3(x)^2 \)[/tex]:
- This function involves a logarithm of [tex]\( x \)[/tex].
- Logarithmic terms do not fit the structure defined for polynomial functions.
- Therefore, [tex]\( p(x) \)[/tex] is not a polynomial function.
From this analysis, we can conclude that the function that is an example of a polynomial function is:
[tex]\[ f(x) = 5x^3 + 2x^2 - 6x + 7 \][/tex]
