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What is the inverse of the function [tex]f(x) = \sqrt{x} + 7[/tex]?

A. [tex]f^{-1}(x) = x^2 - 7[/tex], for [tex]x \geq 7[/tex]

B. [tex]f^{-1}(x) = (x - 7)^2[/tex], for [tex]x \geq 7[/tex]

C. [tex]f^{-1}(x) = x^2 + 7[/tex], for [tex]x \geq -7[/tex]

D. [tex]f^{-1}(x) = (x + 7)^2[/tex], for [tex]x \geq -7[/tex]


Sagot :

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].