Let's break down the steps needed to find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \sqrt{7x - 21} \). Here are the steps in the correct order:
1. Start with the given function and replace \( f(x) \) with \( y \):
[tex]\[
y = \sqrt{7x - 21}
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
y^2 = 7x - 21
\][/tex]
3. Solve for \( x \):
[tex]\[
y^2 + 21 = 7x
\][/tex]
4. Isolate \( x \):
[tex]\[
x = \frac{1}{7} y^2 + 3
\][/tex]
Since we are solving for \( f^{-1}(x) \), we swap \( x \) and \( y \):
5. Replace \( y \) with \( x \) as a last step:
[tex]\[
y = \frac{1}{7} x^2 + 3
\][/tex]
Thus, the steps to find the inverse function \( f^{-1}(x) = \frac{1}{7} x^2 + 3 \) are as follows:
[tex]\[
\begin{array}{c}
x=\sqrt{7 y-21} \\
x^2+21=7 y \\
\frac{1}{7} x^2+3=y \\
\frac{1}{7} x^2+3=f^{-1}(x), \text { where } x \geq 0
\end{array}
\][/tex]
Hence, the correct order of steps is:
1. \( x = \sqrt{7y - 21} \)
2. \( x^2 + 21 = 7y \)
3. \( \frac{1}{7} x^2 + 3 = y \)
4. \( \frac{1}{7} x^2 + 3 = f^{-1}(x), \text{ where } x \geq 0 \)
These steps properly illustrate the process of finding the inverse function.
