To determine whether the function [tex]\( f(x) = 1 - \sqrt{x - 1} \)[/tex] crosses the x-axis and why, let's analyze the function step-by-step.
1. Domain and Initial Point:
- The function [tex]\( f(x) = 1 - \sqrt{x - 1} \)[/tex] is defined for [tex]\( x \geq 1 \)[/tex], because the expression under the square root, [tex]\( x - 1 \)[/tex], must be non-negative.
- At [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1 - \sqrt{1 - 1} = 1 - 0 = 1.
\][/tex]
- Therefore, the function begins at the point [tex]\( (1, 1) \)[/tex].
2. Behavior as [tex]\( x \)[/tex] Increases:
- As [tex]\( x \)[/tex] increases beyond 1, the value [tex]\( \sqrt{x - 1} \)[/tex] also increases.
- Since [tex]\( \sqrt{x - 1} \)[/tex] is subtracted from 1, [tex]\( f(x) \)[/tex] will decrease as [tex]\( x \)[/tex] increases.
3. Determining the x-axis Crossing:
- To find if and where the function crosses the x-axis (i.e., where [tex]\( f(x) = 0 \)[/tex]):
[tex]\[
1 - \sqrt{x - 1} = 0.
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
\sqrt{x - 1} = 1,
\][/tex]
[tex]\[
x - 1 = 1,
\][/tex]
[tex]\[
x = 2.
\][/tex]
- Therefore, the function [tex]\( f(x) = 1 - \sqrt{x - 1} \)[/tex] crosses the x-axis at [tex]\( x = 2 \)[/tex], giving us the point [tex]\( (2, 0) \)[/tex].
4. Conclusion:
- The function starts at [tex]\( (1, 1) \)[/tex] and decreases as [tex]\( x \)[/tex] increases.
- It crosses the x-axis at [tex]\( (2, 0) \)[/tex].
Given the analysis above, we can conclude:
Yes, because it will begin at [tex]\((1,1)\)[/tex] and decrease without bound.
