Question

4
select the correct answer.
consider functions p and q.
p(x) = \log_2(x - 1)
q(x) = 2^x - 1
which statement is true about these functions?
a. the x-intercept of function p is the same as the x-intercept of function q.
b. the x-intercepts cannot be compared because either p or q does not have an x-intercept.
c. the x-intercept of function p is less than the x-intercept of function q.
d. the x-intercept of function p is greater than the x-intercept of function q.

Explanation:

Step1: Find x-intercept of \( p(x) \)

Set \( p(x) = 0 \), so \( \log_2(x - 1) = 0 \). By log property, \( x - 1 = 2^0 = 1 \), so \( x = 2 \).

Step2: Find x-intercept of \( q(x) \)

Set \( q(x) = 0 \), so \( 2^x - 1 = 0 \). Then \( 2^x = 1 \), so \( x = 0 \).

Step3: Compare x-intercepts

\( p(x) \) has x-intercept 2, \( q(x) \) has x-intercept 0. So 2 > 0.

Answer:

D. The x-intercept of function \( p \) is greater than the x-intercept of function \( q \).