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.
