Answer :
To find the inverse of the function \( f(x) = x^2 - 16 \) with the domain \( x \geq 0 \), follow these steps:
1. Express the function in terms of \( y \):
[tex]\[ y = f(x) = x^2 - 16 \][/tex]
2. Solve for \( x \) in terms of \( y \):
[tex]\[ y = x^2 - 16 \][/tex]
Add 16 to both sides:
[tex]\[ y + 16 = x^2 \][/tex]
Take the square root of both sides. Since the domain of \( f(x) \) is \( x \geq 0 \), only the non-negative root is considered:
[tex]\[ x = \sqrt{y + 16} \][/tex]
3. Write the inverse function:
[tex]\[ f^{-1}(x) = \sqrt{x + 16} \][/tex]
4. Check options:
- Option A: \( f^{-1}(x) = \sqrt{x + 16} \) (matches our derived inverse function)
- Option B: \( f^{-1}(x) = \sqrt{x} + 4 \) (does not match)
- Option C: \( f^{-1}(x) = \sqrt{x - 16} \) (does not match)
- Option D: \( f^{-1}(x) = \sqrt{x} - 4 \) (does not match)
From the calculations, the correct answer is:
A. [tex]\( f^{-1}(x) = \sqrt{x + 16} \)[/tex]
1. Express the function in terms of \( y \):
[tex]\[ y = f(x) = x^2 - 16 \][/tex]
2. Solve for \( x \) in terms of \( y \):
[tex]\[ y = x^2 - 16 \][/tex]
Add 16 to both sides:
[tex]\[ y + 16 = x^2 \][/tex]
Take the square root of both sides. Since the domain of \( f(x) \) is \( x \geq 0 \), only the non-negative root is considered:
[tex]\[ x = \sqrt{y + 16} \][/tex]
3. Write the inverse function:
[tex]\[ f^{-1}(x) = \sqrt{x + 16} \][/tex]
4. Check options:
- Option A: \( f^{-1}(x) = \sqrt{x + 16} \) (matches our derived inverse function)
- Option B: \( f^{-1}(x) = \sqrt{x} + 4 \) (does not match)
- Option C: \( f^{-1}(x) = \sqrt{x - 16} \) (does not match)
- Option D: \( f^{-1}(x) = \sqrt{x} - 4 \) (does not match)
From the calculations, the correct answer is:
A. [tex]\( f^{-1}(x) = \sqrt{x + 16} \)[/tex]
