For the function f(x)=9(x+2)^((1)/(7)), find f^(-1)(x). f^(-1)(x)=((x-2)^(7))/(9) f^(-1)(x)=(x^(7)-2)/(9) f^(-1)(x)=((x-2)/(9))^(7) f^(-1)(x)=((x)/(9))^(7)-2

Replace with y: To find the inverse function, we start by replacing f(x) with y for convenience: y = 9(x + 2)^(1/7)Swap x and y: Next, we swap x and y to begin solving for the new y, which will be the inverse function: x = 9(y + 2)^(1/7)Isolate y term: Now, we isolate the term containing y by dividing both sides by 9: (x/9) = (y + 2)^(1/7)Remove exponent: To remove the exponent, we raise both sides of the equation to the power of 7: ((x/9)^7) = y + 2Subtract 2: Subtract 2 from both sides to solve for y: ((x/9)^7) - 2 = yReplace with f^(-1)(x): Finally, we replace y with f^(-1)(x) to denote the inverse function: f^(-1)(x) = ((x/9)^7) - 2

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Q. For the function

f(x)=9(x+2)^{\frac{1}{7}}

, find

f^{-1}(x)

\newline

f^{-1}(x)=\frac{(x-2)^{7}}{9}

\newline

f^{-1}(x)=\frac{x^{7}-2}{9}

\newline

f^{-1}(x)=\left(\frac{x-2}{9}\right)^{7}

\newline

f^{-1}(x)=\left(\frac{x}{9}\right)^{7}-2

Replace with $y$ : To find the inverse function, we start by replacing $f(x)$ with $y$ for convenience: $\newline$ $y = 9(x + 2)^{\frac{1}{7}}$
Swap x and y: Next, we swap x and y to begin solving for the new y, which will be the inverse function: $\newline$ $x = 9(y + 2)^{\frac{1}{7}}$
Isolate y term: Now, we isolate the term containing y by dividing both sides by $\frac{x}{9}$ = $(y + 2)^{\frac{1}{7}}$
Remove exponent: To remove the exponent, we raise both sides of the equation to the power of $7$ : $\left(\frac{x}{9}\right)^7 = y + 2$
Subtract $2$ : Subtract $2$ from both sides to solve for $y$ : $((x/9)^7) - 2 = y$
Replace with $f^{-1}(x)$ : Finally, we replace $y$ with $f^{-1}(x)$ to denote the inverse function: $f^{-1}(x) = \left(\frac{x}{9}\right)^7 - 2$