To determine which of the given functions is an example of an exponential function, we need to recall the definition of an exponential function. An exponential function is a mathematical function of the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( b \)[/tex] is positive and not equal to 1.
Here are the given functions:
1. [tex]\( f(x) = 4x^3 + 2x^2 - 3x + 9 \)[/tex]
2. [tex]\( g(x) = \log_2(x)^3 \)[/tex]
3. [tex]\( h(x) = \frac{x^2 + 2x - 7}{x + 1} \)[/tex]
4. [tex]\( p(x) = 500(1.02)^x \)[/tex]
Let's analyze each function:
1. [tex]\( f(x) = 4x^3 + 2x^2 - 3x + 9 \)[/tex]
- This is a polynomial function because it is composed of terms with [tex]\( x \)[/tex] raised to non-negative integer powers. This is not an exponential function.
2. [tex]\( g(x) = \log_2(x)^3 \)[/tex]
- This function involves the logarithm of [tex]\( x \)[/tex] raised to a power. Specifically, the logarithm is raised to the third power. Logarithmic functions are not exponential functions. This is not an exponential function.
3. [tex]\( h(x) = \frac{x^2 + 2x - 7}{x + 1} \)[/tex]
- This is a rational function because it is the ratio of two polynomials. Rational functions are not exponential functions. This is not an exponential function.
4. [tex]\( p(x) = 500(1.02)^x \)[/tex]
- This function matches the form of an exponential function, [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a = 500 \)[/tex] and [tex]\( b = 1.02 \)[/tex]. Both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( b = 1.02 > 0\)[/tex]. This is indeed an exponential function.
Based on this analysis, we can conclude that [tex]\( p(x) = 500(1.02)^x \)[/tex] is the only function that fits the definition of an exponential function.
Therefore, the answer is [tex]\( \boxed{4} \)[/tex].
