To determine the coordinates of the points on the inverse of the function [tex]\( f(x) = x^{\frac{1}{2}} \)[/tex], we need to first understand the relationship between a function and its inverse. For a function [tex]\( f \)[/tex] and its inverse [tex]\( f^{-1} \)[/tex]:
[tex]\[ f(a) = b \implies f^{-1}(b) = a \][/tex]
The given points on [tex]\( f(x) = x^{\frac{1}{2}} \)[/tex] are:
1. [tex]\((0, 0)\)[/tex]
2. [tex]\((1, 1)\)[/tex]
3. [tex]\((4, 2)\)[/tex]
To find the corresponding points on the inverse function [tex]\( f^{-1}(x) \)[/tex], we simply swap the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates of each point. This is because if [tex]\( (a, b) \)[/tex] is a point on the function [tex]\( f \)[/tex], then [tex]\( (b, a) \)[/tex] will be a point on [tex]\( f^{-1} \)[/tex].
Now, let's identify the coordinates on [tex]\( f^{-1}(x) \)[/tex]:
1. For [tex]\((0, 0)\)[/tex]:
- Swapping coordinates gives [tex]\((0, 0)\)[/tex]
2. For [tex]\((1, 1)\)[/tex]:
- Swapping coordinates gives [tex]\((1, 1)\)[/tex]
3. For [tex]\((4, 2)\)[/tex]:
- Swapping coordinates gives [tex]\((2, 4)\)[/tex]
Therefore, the points on the inverse function [tex]\( f^{-1}(x) \)[/tex] corresponding to the given points are:
- [tex]\((0, 0)\)[/tex]
- [tex]\((1, 1)\)[/tex]
- [tex]\((2, 4)\)[/tex]
So, the named coordinates corresponding to the points provided are:
a. [tex]\((1,1)\)[/tex]
c. [tex]\((0,0)\)[/tex]
e. [tex]\((2,4)\)[/tex]
