To determine which exponential function has an [tex]\(x\)[/tex]-intercept, we need to find the [tex]\(x\)[/tex]-intercept for each function. An [tex]\(x\)[/tex]-intercept is a point where the function value [tex]\(f(x)\)[/tex] is equal to 0. Let's analyze each function one by one:
### Function A: [tex]\(f(x) = 100^{x-5} - 1\)[/tex]
To find the [tex]\(x\)[/tex]-intercept, set [tex]\(f(x) = 0\)[/tex]:
[tex]\[
100^{x-5} - 1 = 0
\][/tex]
[tex]\[
100^{x-5} = 1
\][/tex]
For [tex]\(100^{x-5}\)[/tex] to be equal to 1, the exponent must be 0 because any positive base raised to the power of 0 is 1:
[tex]\[
x - 5 = 0
\][/tex]
[tex]\[
x = 5
\][/tex]
Therefore, function A has an [tex]\(x\)[/tex]-intercept at [tex]\(x = 5\)[/tex].
### Function B: [tex]\(f(x) = 3^{x-4} + 2\)[/tex]
To find the [tex]\(x\)[/tex]-intercept, set [tex]\(f(x) = 0\)[/tex]:
[tex]\[
3^{x-4} + 2 = 0
\][/tex]
[tex]\[
3^{x-4} = -2
\][/tex]
This equation has no solution because [tex]\(3^{x-4}\)[/tex] is always positive for any real number [tex]\(x\)[/tex] (an exponential function with a positive base cannot yield a negative value).
### Function C: [tex]\(f(x) = 7^{x-1} + 1\)[/tex]
To find the [tex]\(x\)[/tex]-intercept, set [tex]\(f(x) = 0\)[/tex]:
[tex]\[
7^{x-1} + 1 = 0
\][/tex]
[tex]\[
7^{x-1} = -1
\][/tex]
This equation has no solution because [tex]\(7^{x-1}\)[/tex] is always positive for any real number [tex]\(x\)[/tex] (similarly, an exponential function with a positive base cannot yield a negative value).
### Function D: [tex]\(f(x) = -8^{x+1} - 3\)[/tex]
To find the [tex]\(x\)[/tex]-intercept, set [tex]\(f(x) = 0\)[/tex]:
[tex]\[
-8^{x+1} - 3 = 0
\][/tex]
[tex]\[
-8^{x+1} = 3
\][/tex]
[tex]\[
8^{x+1} = -3
\][/tex]
This equation has no solution because [tex]\(8^{x+1}\)[/tex] is always positive for any real number [tex]\(x\)[/tex] (likewise, an exponential function with a positive base cannot yield a negative value).
After evaluating each function, we see that only Function A: [tex]\(f(x) = 100^{x-5} - 1\)[/tex] has an [tex]\(x\)[/tex]-intercept. Therefore, the correct answer is:
A. [tex]\(f(x)=100^{x-5}-1\)[/tex]
