Question

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 \( f(x) = \sqrt{x^2 + 13} \), let's follow the steps correctly and fill in the blanks:

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

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

3. Now, isolate \( x^2 \) by subtracting 13 from both sides:
\[
x^2 = y^2 - 13
\]

4. Taking the square root of both sides gives:
\[
x = \pm \sqrt{y^2 - 13}
\]

5. Since we are looking for an inverse function, we can express \( y \) in terms of \( x \):
\[
y = \sqrt{x^2 - 13}
\]

Now we can summarize the completed steps:

1. \( y = \sqrt{x^2 + 13} \)
2. \( y^2 = x^2 + 13 \)
3. \( x^2 = y^2 - 13 \)
4. \( x = \sqrt{y^2 - 13} \) (Taking the positive root for the inverse)
5. Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt{x^2 - 13}
\]

Final Statement:
- The correct fill-in for the blanks is:
- \( x^2 - 13 = y \)
- \( f^{-1}(x) = \sqrt{x^2 - 13} \)

(Note: Ensure that when defining the inverse, you consider the appropriate domain/range based on the original function to maintain its properties.)