To understand the relationship between the graphs of the functions [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) = \sqrt{x-1} \)[/tex], we need to compare them carefully.
### Relationship Between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
Given:
[tex]\[ f(x) = \sqrt{x} \][/tex]
[tex]\[ g(x) = \sqrt{x-1} \][/tex]
The function [tex]\( g(x) \)[/tex] is derived by substituting [tex]\( x-1 \)[/tex] in place of [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]. This substitution implies a horizontal shift in the graph of the function. Specifically:
- The graph of [tex]\( g(x) \)[/tex] is shifted 1 unit to the right of the graph of [tex]\( f(x) \)[/tex].
### Finding the Corresponding Point
We are given that [tex]\( (1,1) \)[/tex] is a point on the graph of [tex]\( f(x) \)[/tex]. This means:
[tex]\[ f(1) = \sqrt{1} = 1 \][/tex]
To find the corresponding point on the graph of [tex]\( g(x) \)[/tex], we use the fact that the graph of [tex]\( g(x) \)[/tex] is the horizontal right shift of [tex]\( f(x) \)[/tex] by 1 unit.
So if [tex]\( (1,1) \)[/tex] is a point on [tex]\( f(x) \)[/tex]:
- We increase the x-coordinate by 1 to find the corresponding x-coordinate on [tex]\( g(x) \)[/tex].
Therefore, the corresponding x-value for the point on [tex]\( g(x) \)[/tex] is:
[tex]\[ x_{g} = x_{f} + 1 = 1 + 1 = 2 \][/tex]
The y-coordinate stays the same because the function values at these corresponding x-values are equivalent due to the nature of the shift:
[tex]\[ y_{f} = y_{g} \][/tex]
[tex]\[ y_{g} = 1 \][/tex]
So the corresponding point on [tex]\( g(x) \)[/tex] will be [tex]\( (x_{g}, y_{g}) \)[/tex]:
[tex]\[ (2, 1) \][/tex]
### Conclusion
When [tex]\( (1,1) \)[/tex] is a point on the graph of [tex]\( f(x) \)[/tex], the corresponding point on the graph of [tex]\( g(x) \)[/tex] is:
[tex]\[ (2, 1) \][/tex]
So, the blank spaces should be filled as follows:
If [tex]\( (1,1) \)[/tex] is a point of [tex]\( f(x) \)[/tex], then [tex]\( (2, 1) \)[/tex] is the corresponding point of [tex]\( g(x) \)[/tex].
