Answer :
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.
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.
