To determine which exponential function has an [tex]\( x \)[/tex]-intercept, we need to find for which functions [tex]\( f(x) = 0 \)[/tex] has a real solution for [tex]\( x \)[/tex].
Let's analyze each function one by one:
Function A: [tex]\( f(x) = 100^{x-5} - 1 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
100^{x-5} - 1 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
100^{x-5} = 1
\][/tex]
The equation [tex]\( 100^{x-5} = 1 \)[/tex] holds true if [tex]\( 100^{x-5} = 100^0 \)[/tex], which happens when:
[tex]\[
x-5 = 0 \implies x = 5
\][/tex]
Thus, function A has a real [tex]\( x \)[/tex]-intercept at [tex]\( x = 5 \)[/tex].
Function B: [tex]\( f(x) = 3^{x-4} + 2 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
3^{x-4} + 2 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
3^{x-4} = -2
\][/tex]
The base of the exponent, 3, is a positive number, and any power of a positive number remains positive. Hence, [tex]\( 3^{x-4} = -2 \)[/tex] has no real solution because a positive number can never equal a negative number.
Thus, function B has no real [tex]\( x \)[/tex]-intercept.
Function C: [tex]\( f(x) = 7^{x-1} + 1 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
7^{x-1} + 1 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
7^{x-1} = -1
\][/tex]
Similarly to function B, the base 7 is positive, and powers of positive numbers cannot equal negative numbers.
Thus, function C has no real [tex]\( x \)[/tex]-intercept.
Function D: [tex]\( f(x) = -8^{x+1} - 3 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
-8^{x+1} - 3 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
-8^{x+1} = 3
\][/tex]
Here, [tex]\( -8^{x+1} \)[/tex] indicates a negative base raised to a power, which will always yield negative results, and therefore cannot be equal to a positive number.
Thus, function D has no real [tex]\( x \)[/tex]-intercept.
Conclusion:
By examining all functions, we see that the only function which has a real [tex]\( x \)[/tex]-intercept is function A, [tex]\( f(x) = 100^{x-5} - 1 \)[/tex], at [tex]\( x = 5 \)[/tex].
So, the correct answer is A.
