Question

2
select the correct answer.
what is the inverse of function f?
f(x) = √x + 7
a. f⁻¹(x) = x² − 7, for x ≥ 7
b. f⁻¹(x) = x² + 7, for x ≥ -7
c. f⁻¹(x) = (x + 7)², for x ≥ -7
d. f⁻¹(x) = (x − 7)², for x ≥ 7

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = \sqrt{x} + 7 \). This is a standard first step in finding the inverse of a function, as it helps us switch the roles of \( x \) and \( y \) later.

Step2: Swap \( x \) and \( y \)

Now, we interchange \( x \) and \( y \) to get \( x = \sqrt{y} + 7 \). This is based on the definition of an inverse function, where the input and output of the original function are swapped in the inverse function.

Step3: Solve for \( y \)

First, we isolate the square root term. Subtract 7 from both sides of the equation: \( x - 7 = \sqrt{y} \). Then, to eliminate the square root, we square both sides of the equation. Squaring the left side gives \( (x - 7)^2 \), and squaring the right side gives \( y \) (since \( (\sqrt{y})^2 = y \) for \( y \geq 0 \)). So we have \( y = (x - 7)^2 \).

Step4: Determine the domain of the inverse function

The original function \( f(x)=\sqrt{x}+7 \) has a domain of \( x \geq 0 \) (because we can't take the square root of a negative number in the real - number system) and a range of \( y \geq 7 \) (since \( \sqrt{x}\geq0 \), so \( \sqrt{x}+7\geq7 \)). For the inverse function, the domain of the inverse function is the range of the original function. So the domain of \( f^{-1}(x) \) is \( x \geq 7 \). And the inverse function is \( f^{-1}(x)=(x - 7)^2 \) for \( x\geq7 \).

Answer:

D. \( f^{-1}(x)=(x - 7)^2 \), for \( x\geq7 \)