Answer :
To find the inverse of the given function \( f(x) = \sqrt{x} + 7 \), we need to follow these steps:
1. Replace \( f(x) \) with \( y \):
[tex]\[ y = \sqrt{x} + 7 \][/tex]
2. Solve for \( x \) in terms of \( y \):
- Begin by isolating the square root term:
[tex]\[ y - 7 = \sqrt{x} \][/tex]
- Next, square both sides to eliminate the square root:
[tex]\[ (y - 7)^2 = x \][/tex]
3. Express \( x \) as a function of \( y \):
[tex]\[ x = (y - 7)^2 \][/tex]
This means that the inverse function \( f^{-1}(x) \) is:
[tex]\[ f^{-1}(x) = (x - 7)^2 \][/tex]
4. Consider the domain: Since \( f(x) = \sqrt{x} + 7 \) is defined for \( x \geq 0 \), the inverse function \( f^{-1}(x) \) will be defined for \( x \geq 7 \), ensuring that the argument of the square root is non-negative.
Based on this step-by-step solution, the correct answer is:
B. [tex]\( f^{-1}(x) = (x - 7)^2 \)[/tex], for [tex]\( x \geq 7 \)[/tex].
1. Replace \( f(x) \) with \( y \):
[tex]\[ y = \sqrt{x} + 7 \][/tex]
2. Solve for \( x \) in terms of \( y \):
- Begin by isolating the square root term:
[tex]\[ y - 7 = \sqrt{x} \][/tex]
- Next, square both sides to eliminate the square root:
[tex]\[ (y - 7)^2 = x \][/tex]
3. Express \( x \) as a function of \( y \):
[tex]\[ x = (y - 7)^2 \][/tex]
This means that the inverse function \( f^{-1}(x) \) is:
[tex]\[ f^{-1}(x) = (x - 7)^2 \][/tex]
4. Consider the domain: Since \( f(x) = \sqrt{x} + 7 \) is defined for \( x \geq 0 \), the inverse function \( f^{-1}(x) \) will be defined for \( x \geq 7 \), ensuring that the argument of the square root is non-negative.
Based on this step-by-step solution, the correct answer is:
B. [tex]\( f^{-1}(x) = (x - 7)^2 \)[/tex], for [tex]\( x \geq 7 \)[/tex].
