To determine which of the given functions represents an exponential function that passes through the point [tex]\((2,80)\)[/tex], we need to evaluate each function at [tex]\(x = 2\)[/tex] and see which one yields [tex]\(80\)[/tex].
Let's analyze each of the given functions:
1. [tex]\( f(x) = 4 \cdot x^5 \)[/tex]
[tex]\[
f(2) = 4 \cdot (2)^5 = 4 \cdot 32 = 128
\][/tex]
Since [tex]\(128 \neq 80\)[/tex], this function does not pass through [tex]\((2,80)\)[/tex].
2. [tex]\( f(x) = 5 \cdot x^4 \)[/tex]
[tex]\[
f(2) = 5 \cdot (2)^4 = 5 \cdot 16 = 80
\][/tex]
Since [tex]\(80 = 80\)[/tex], this function does pass through [tex]\((2,80)\)[/tex].
3. [tex]\( f(x) = 4 \cdot 5^x \)[/tex]
[tex]\[
f(2) = 4 \cdot 5^2 = 4 \cdot 25 = 100
\][/tex]
Since [tex]\(100 \neq 80\)[/tex], this function does not pass through [tex]\((2,80)\)[/tex].
4. [tex]\( f(x) = 5 \cdot 4^x \)[/tex]
[tex]\[
f(2) = 5 \cdot 4^2 = 5 \cdot 16 = 80
\][/tex]
Since [tex]\(80 = 80\)[/tex], this function also passes through [tex]\((2,80)\)[/tex].
So, there are two functions that pass through the point [tex]\((2,80)\)[/tex]:
- [tex]\(f(x) = 5 \cdot x^4\)[/tex]
- [tex]\(f(x) = 5 \cdot 4^x\)[/tex]
Considering the question asks for an exponential function, we should select the function of the exponential form among these choices. The function [tex]\(f(x) = 5 \cdot x^4\)[/tex] is a polynomial function, not an exponential function. The function [tex]\(f(x) = 5 \cdot 4^x\)[/tex] is an exponential function.
Thus, the correct function is:
[tex]\[ f(x) = 5 \cdot 4^x \][/tex]
So, the exponential function that passes through the point [tex]\((2,80)\)[/tex] is:
[tex]\[ f(x) = 5 \cdot 4^x \][/tex]
Therefore, the correct equation from the options provided is:
[tex]\[ f(x) = 5(4)^x \][/tex]
