To determine which exponential function does not have an [tex]$x$[/tex]-intercept, we analyze each function to see if and where they cross the [tex]$x$[/tex]-axis, that is, where [tex]\(f(x) = 0\)[/tex].
### Option A: [tex]\( f(x) = 5^{x-5} - 1 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
5^{x-5} - 1 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
5^{x-5} = 1
\][/tex]
3. Since [tex]\( 5^0 = 1 \)[/tex], we get:
[tex]\[
x - 5 = 0 \implies x = 5
\][/tex]
This function has an x-intercept at [tex]\( x = 5 \)[/tex].
### Option B: [tex]\( f(x) = 5^{x-5} - 5 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
5^{x-5} - 5 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
5^{x-5} = 5
\][/tex]
3. Since [tex]\( 5^1 = 5 \)[/tex], we get:
[tex]\[
x - 5 = 1 \implies x = 6
\][/tex]
This function has an x-intercept at [tex]\( x = 6 \)[/tex].
### Option C: [tex]\( f(x) = -5^{x-5} + 5 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
-5^{x-5} + 5 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
-5^{x-5} = -5
\][/tex]
3. Multiplying both sides by -1, we get:
[tex]\[
5^{x-5} = 5
\][/tex]
4. Since [tex]\( 5^1 = 5 \)[/tex], we get:
[tex]\[
x - 5 = 1 \implies x = 6
\][/tex]
This function has an x-intercept at [tex]\( x = 6 \)[/tex].
### Option D: [tex]\( f(x) = -5^{x-5} - 1 \)[/tex]
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
-5^{x-5} - 1 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
-5^{x-5} = 1
\][/tex]
3. Since the term [tex]\(-5^{x-5}\)[/tex] is always negative (as the base 5 to any real power gives a positive number, and the negative sign makes it negative), it can never equal 1, which is positive.
Therefore, this function does not have an x-intercept.
### Conclusion:
The only function that does not cross the x-axis is:
[tex]\[ \boxed{D. \: f(x) = -5^{x-5} - 1} \][/tex]
