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 check if the result is [tex]\(y = 80\)[/tex].
Here are the given functions:
1. [tex]\(f(x) = 4 \cdot x^5\)[/tex]
2. [tex]\(f(x) = 5 \cdot x^4\)[/tex]
3. [tex]\(f(x) = 4 \cdot 5^x\)[/tex]
4. [tex]\(f(x) = 5 \cdot 4^x\)[/tex]
Let's evaluate each function at [tex]\(x = 2\)[/tex]:
1. Function 1: [tex]\(f(x) = 4 \cdot x^5\)[/tex]
[tex]\[
f(2) = 4 \cdot (2)^5 = 4 \cdot 32 = 128
\][/tex]
[tex]\[
f(2) \neq 80
\][/tex]
This function does not pass through the point [tex]\((2, 80)\)[/tex].
2. Function 2: [tex]\(f(x) = 5 \cdot x^4\)[/tex]
[tex]\[
f(2) = 5 \cdot (2)^4 = 5 \cdot 16 = 80
\][/tex]
[tex]\[
f(2) = 80
\][/tex]
This function passes through the point [tex]\((2, 80)\)[/tex].
3. Function 3: [tex]\(f(x) = 4 \cdot 5^x\)[/tex]
[tex]\[
f(2) = 4 \cdot (5)^x = 4 \cdot 25 = 100
\][/tex]
[tex]\[
f(2) \neq 80
\][/tex]
This function does not pass through the point [tex]\((2, 80)\)[/tex].
4. Function 4: [tex]\(f(x) = 5 \cdot 4^x\)[/tex]
[tex]\[
f(2) = 5 \cdot (4)^2 = 5 \cdot 16 = 80
\][/tex]
[tex]\[
f(2) = 80
\][/tex]
This function passes through the point [tex]\((2, 80)\)[/tex].
From our evaluations, the functions that pass through the point [tex]\((2, 80)\)[/tex] are:
- [tex]\(f(x) = 5 \cdot x^4\)[/tex]
- [tex]\(f(x) = 5 \cdot 4^x\)[/tex]
However, since the question is asking for an exponential function that passes through [tex]\((2, 80)\)[/tex], we only consider the appropriate type.
Therefore, the correct exponential function is:
[tex]\[ f(x) = 5 \cdot 4^x \][/tex]
[tex]\(\boxed{4}\)[/tex]
