For the function f(x)=(x^((1)/(7))+6)/(4), find f^(-1)(x). f^(-1)(x)=(4x)^(7)-6 f^(-1)(x)=(4x-6)^(7) f^(-1)(x)=(4(x-6))^(7) f^(-1)(x)=4(x-6)^(7)

Write function as y: To find the inverse function, we first write the function as y: y = (x^(1/7) + 6) / 4Swap x and y: Next, we swap x and y to begin solving for the inverse function: x = (y^(1/7) + 6) / 4Eliminate denominator: Multiply both sides by 4 to eliminate the denominator: 4x = y^(1/7) + 6Isolate term with y: Subtract 6 from both sides to isolate the term with y: 4x - 6 = y^(1/7)Eliminate exponent on y: Raise both sides to the power of 7 to eliminate the exponent on y: (4x - 6)^7 = (y^(1/7))^7Simplify to get y: Simplify the right side to get y by itself: (4x - 6)^7 = yInverse function: Now we have the inverse function: f^(-1)(x) = (4x - 6)^7

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Find derivatives of using multiple formulae

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Q. For the function

f(x)=\frac{x^{\frac{1}{7}}+6}{4}

, find

f^{-1}(x)

\newline

f^{-1}(x)=(4 x)^{7}-6

\newline

f^{-1}(x)=(4 x-6)^{7}

\newline

f^{-1}(x)=(4(x-6))^{7}

\newline

f^{-1}(x)=4(x-6)^{7}

Write function as $y$ : To find the inverse function, we first write the function as $y$ : $y = \frac{{(x^{\frac{1}{7}} + 6)}}{4}$
Swap x and y: Next, we swap x and y to begin solving for the inverse function: $\newline$ $x = \frac{y^{\frac{1}{7}} + 6}{4}$
Eliminate denominator: Multiply both sides by $4$ to eliminate the denominator: $\newline$ $4x = y^{\frac{1}{7}} + 6$
Isolate term with y: Subtract $6$ from both sides to isolate the term with y: $\newline$ $4x - 6 = y^{\frac{1}{7}}$
Eliminate exponent on y: Raise both sides to the power of $7$ to eliminate the exponent on y: $(4x - 6)^7 = (y^{\frac{1}{7}})^7$
Simplify to get $y$ : Simplify the right side to get $y$ by itself: $(4x - 6)^7 = y$
Inverse function: Now we have the inverse function: $f^{-1}(x) = (4x - 6)^7$