Answer :
Let's find the inverse of the function [tex]\( f(x) = \sqrt{x} + 7 \)[/tex] by following these steps:
1. Start with the given function:
[tex]\[ f(x) = \sqrt{x} + 7 \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{x} + 7 \][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y - 7 = \sqrt{x} \][/tex]
4. Square both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ (y - 7)^2 = x \][/tex]
5. Rewrite the expression for [tex]\( x \)[/tex]:
[tex]\[ x = (y - 7)^2 \][/tex]
6. Now switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = (x - 7)^2 \][/tex]
7. Determine the domain of the inverse function:
Since the original function [tex]\( f(x) = \sqrt{x} + 7 \)[/tex] takes non-negative values of [tex]\( x \)[/tex] and adds 7 (which shifts the graph upwards by 7 units), [tex]\( y \)[/tex] must be at least 7 for the inverse to be valid (to ensure the output of the inverse function remains in the domain of the original function).
So, the inverse function is:
[tex]\[ f^{-1}(x) = (x - 7)^2, \text{ for } x \geq 7 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2 \text{ (B). } f^{-1}(x) = (x - 7)^2, \text{ for } x \geq 7} \][/tex]
1. Start with the given function:
[tex]\[ f(x) = \sqrt{x} + 7 \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{x} + 7 \][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y - 7 = \sqrt{x} \][/tex]
4. Square both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ (y - 7)^2 = x \][/tex]
5. Rewrite the expression for [tex]\( x \)[/tex]:
[tex]\[ x = (y - 7)^2 \][/tex]
6. Now switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = (x - 7)^2 \][/tex]
7. Determine the domain of the inverse function:
Since the original function [tex]\( f(x) = \sqrt{x} + 7 \)[/tex] takes non-negative values of [tex]\( x \)[/tex] and adds 7 (which shifts the graph upwards by 7 units), [tex]\( y \)[/tex] must be at least 7 for the inverse to be valid (to ensure the output of the inverse function remains in the domain of the original function).
So, the inverse function is:
[tex]\[ f^{-1}(x) = (x - 7)^2, \text{ for } x \geq 7 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2 \text{ (B). } f^{-1}(x) = (x - 7)^2, \text{ for } x \geq 7} \][/tex]
