Question

Fill in the blanks to complete the steps to find the inverse of f(x)=x2+13−−−−−−√
(4 points)
y=x2+13−−−−−−√

=y2+13−−−−−−√

=y2+13

x2−13=

=y

f−1(x)=2(x2−13)

Answer 1

To find the inverse of the function \( f(x) = \sqrt{x^2 + 13} \), we can follow these steps and fill in the blanks:

1. Start with the function:
\[
y = \sqrt{x^2 + 13}
\]

2. Square both sides to eliminate the square root:
\[
y^2 = x^2 + 13
\]

3. Rearrange the equation to solve for \( x^2 \):
\[
x^2 = y^2 - 13
\]

4. Finally, take the square root of both sides to solve for \( x \):
\[
x = \sqrt{y^2 - 13} \quad \text{(assuming } x \text{ is non-negative)}
\]

5. Switch \( x \) and \( y \) to find the inverse:
\[
f^{-1}(x) = \sqrt{x^2 - 13}
\]

So, the complete steps to find the inverse of \( f(x) = \sqrt{x^2 + 13} \) are:

1. \( y = \sqrt{x^2 + 13} \)

2. \( y^2 = x^2 + 13 \)

3. \( x^2 = y^2 - 13 \)

4. \( x = \sqrt{y^2 - 13} \)

5. \( f^{-1}(x) = \sqrt{x^2 - 13} \)

The equation you provided for the inverse, \( f^{-1}(x) = 2(x^2 - 13) \), was not correct for the inverse function derived from \( f(x) \).

The correct inverse function is:
\[
f^{-1}(x) = \sqrt{x^2 - 13}
\] (assuming \( x \) is sufficiently large so that the expression under the square root is non-negative).