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Consider the function [tex]f[/tex].
[tex]f(x) = \sqrt{7x - 21}[/tex]

Place the steps for finding [tex]f^{-1}(x)[/tex] in the correct order.
[tex]
\begin{array}{c}
y = \sqrt{7x - 21} \\
x = \sqrt{7y - 21} \\
x^2 + 21 = 7y \\
x^2 = 7y - 21 \\
\frac{1}{7}(x^2 - 21) = f^{-1}(x), \text{ where } x \geq 0
\end{array}
[/tex]



Answer :

To find the inverse of the function [tex]\( f(x) = \sqrt{7x - 21} \)[/tex], we need to go through a sequence of steps methodically. Follow the steps below to determine the inverse function [tex]\( f^{-1}(x) \)[/tex]:

1. Start with the original function:
[tex]\[ y = \sqrt{7x - 21} \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt{7y - 21} \][/tex]

3. Square both sides to eliminate the square root:
[tex]\[ x^2 = 7y - 21 \][/tex]

4. Solve for [tex]\( y \)[/tex]:
[tex]\[ x^2 + 21 = 7y \][/tex]
[tex]\[ y = \frac{1}{7}(x^2 + 21) \][/tex]

5. Rewrite the inverse function as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{1}{7}(x^2 + 21) \][/tex]

Here are the correct steps placed in order:

1. [tex]\[ y = \sqrt{7x - 21} \][/tex]

2. [tex]\[ x = \sqrt{7 y - 21} \][/tex]

3. [tex]\[ x^2 = 7 y - 21 \][/tex]

4. [tex]\[ x^2 + 21 = 7 y \][/tex]

5. [tex]\[ f^{-1}(x) = \frac{1}{7} (x^2 + 21) \quad \text{where} \quad x \geq 0 \][/tex]

Thus, the final correct order is:
1. [tex]\( y = \sqrt{7 x - 21} \)[/tex]
2. [tex]\( x = \sqrt{7 y - 21} \)[/tex]
3. [tex]\( x^2 = 7 y - 21 \)[/tex]
4. [tex]\( x^2 + 21 = 7 y \)[/tex]
5. [tex]\( \frac{1}{7} (x^2 + 21) = f^{-1}(x) \quad \text{where} \quad x \geq 0 \)[/tex]

Ensure to recognize that the domain restriction [tex]\( x \geq 0 \)[/tex] is necessary since we started with a square root function, which is defined for non-negative values.