To determine which of the given functions passes through the point [tex]\((2, 80)\)[/tex], we need to evaluate each function at [tex]\( x = 2 \)[/tex] and check if the output matches [tex]\( y = 80 \)[/tex].
Let's evaluate each function step-by-step:
1. Function [tex]\( f(x) = 4(x)^5 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 4 \cdot (2)^5
\][/tex]
- Calculate [tex]\( 2^5 \)[/tex]:
[tex]\[
2^5 = 32
\][/tex]
- Then multiply by 4:
[tex]\[
4 \cdot 32 = 128
\][/tex]
- Here, [tex]\( f(2) = 128 \)[/tex], which does not equal 80. So this function does not pass through [tex]\((2, 80)\)[/tex].
2. Function [tex]\( f(x) = 5(x)^4 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 5 \cdot (2)^4
\][/tex]
- Calculate [tex]\( 2^4 \)[/tex]:
[tex]\[
2^4 = 16
\][/tex]
- Then multiply by 5:
[tex]\[
5 \cdot 16 = 80
\][/tex]
- Here, [tex]\( f(2) = 80 \)[/tex], which matches the point [tex]\((2, 80)\)[/tex]. Therefore, this function passes through [tex]\((2, 80)\)[/tex].
3. Function [tex]\( f(x) = 4(5)^x \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 4 \cdot (5)^2
\][/tex]
- Calculate [tex]\( 5^2 \)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
- Then multiply by 4:
[tex]\[
4 \cdot 25 = 100
\][/tex]
- Here, [tex]\( f(2) = 100 \)[/tex], which does not equal 80. So this function does not pass through [tex]\((2, 80)\)[/tex].
4. Function [tex]\( f(x) = 5(4)^x \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 5 \cdot (4)^2
\][/tex]
- Calculate [tex]\( 4^2 \)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
- Then multiply by 5:
[tex]\[
5 \cdot 16 = 80
\][/tex]
- Here, [tex]\( f(2) = 80 \)[/tex], which matches the point [tex]\((2, 80)\)[/tex]. Therefore, this function also passes through [tex]\((2, 80)\)[/tex].
Both [tex]\( f(x) = 5(x)^4 \)[/tex] and [tex]\( f(x) = 5(4)^x \)[/tex] pass through the point [tex]\((2, 80)\)[/tex]. Given the solution requested one of the matching functions, one might choose either. Let's select the first matching function found:
Thus, the function [tex]\( f(x) = 5(x)^4 \)[/tex] represents an exponential function that passes through the point [tex]\((2, 80)\)[/tex].
