To solve this problem step-by-step, we need to find the [tex]$x$[/tex]-intercepts of both functions [tex]\( p(x) \)[/tex] and [tex]\( Q(x) \)[/tex], and compare them.
### Step 1: Find the [tex]$x$[/tex]-intercept of function [tex]\( p(x) \)[/tex]
The [tex]$x$[/tex]-intercept of a function is the value of [tex]\( x \)[/tex] for which the function equals zero. For [tex]\( p(x) \)[/tex]:
[tex]\[ p(x) = \log_2(x-1) \][/tex]
Set [tex]\( p(x) = 0 \)[/tex]:
[tex]\[ \log_2(x-1) = 0 \][/tex]
To solve for [tex]\( x \)[/tex], exponentiate both sides using base 2:
[tex]\[ x - 1 = 2^0 \][/tex]
[tex]\[ x - 1 = 1 \][/tex]
[tex]\[ x = 2 \][/tex]
So, the [tex]$x$[/tex]-intercept of [tex]\( p(x) \)[/tex] is [tex]\( x = 2 \)[/tex].
### Step 2: Find the [tex]$x$[/tex]-intercept of function [tex]\( Q(x) \)[/tex]
Similarly, we find the [tex]$x$[/tex]-intercept for [tex]\( Q(x) \)[/tex] by setting the function equal to zero:
[tex]\[ Q(x) = 2^x - 1 \][/tex]
Set [tex]\( Q(x) = 0 \)[/tex]:
[tex]\[ 2^x - 1 = 0 \][/tex]
To solve for [tex]\( x \)[/tex]:
[tex]\[ 2^x = 1 \][/tex]
Since [tex]\( 2^0 = 1 \)[/tex]:
[tex]\[ x = 0 \][/tex]
So, the [tex]$x$[/tex]-intercept of [tex]\( Q(x) \)[/tex] is [tex]\( x = 0 \)[/tex].
### Step 3: Compare the [tex]$x$[/tex]-intercepts
We now compare the intercepts:
- [tex]\( x \)[/tex]-intercept of [tex]\( p(x) \)[/tex] is [tex]\( x = 2 \)[/tex]
- [tex]\( x \)[/tex]-intercept of [tex]\( Q(x) \)[/tex] is [tex]\( x = 0 \)[/tex]
Clearly, [tex]\( 2 \)[/tex] is greater than [tex]\( 0 \)[/tex].
### Conclusion
Thus, the [tex]$x$[/tex]-intercept of function [tex]\( p(x) \)[/tex] is greater than the [tex]$x$[/tex]-intercept of function [tex]\( Q(x) \)[/tex].
The correct answer is:
[tex]\[ \boxed{D} \][/tex]
