The function [tex]\( f(x) = 6^x \)[/tex] is an exponential function. Let's analyze it step by step:
1. Base of the exponential function:
The function is given by [tex]\( f(x) = 6^x \)[/tex]. Here, the base of the exponential function is 6. So, the base is 6.
2. Calculating [tex]\( f(-2) \)[/tex]:
The value of the function at [tex]\( x = -2 \)[/tex] is found by substituting [tex]\(-2\)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(-2) = 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \approx 0.027777777777777776
\][/tex]
3. Calculating [tex]\( f(0) \)[/tex]:
The value of the function at [tex]\( x = 0 \)[/tex] is found by substituting [tex]\(0\)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(0) = 6^0 = 1
\][/tex]
4. Calculating [tex]\( f(2) \)[/tex]:
The value of the function at [tex]\( x = 2 \)[/tex] is found by substituting [tex]\(2\)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(2) = 6^2 = 36
\][/tex]
5. Calculating [tex]\( f(6) \)[/tex]:
The value of the function at [tex]\( x = 6 \)[/tex] is found by substituting [tex]\(6\)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(6) = 6^6 = 46656
\][/tex]
By summarizing these calculations, we get:
[tex]\[
f(x) = 6^x \text{ is an exponential function with base } 6 \, ; \, f(-2) = 0.027777777777777776, \, f(0) = 1, \, f(2) = 36, \, \text{and} \, f(6) = 46656
\][/tex]
So, the completed statement is:
The function [tex]\( f(x)=6^x \)[/tex] is an exponential function with base [tex]\( 6 \)[/tex]; [tex]\( f(-2) = 0.027777777777777776 \)[/tex], [tex]\( f(0) = 1 \)[/tex], [tex]\( f(2) = 36 \)[/tex], and [tex]\( f(6) = 46656 \)[/tex].
