For the function f(x)=(root(7)(x))/(5)+8, find f^(-1)(x). f^(-1)(x)=5x^(7)-8 f^(-1)(x)=(5x)^(7)-8 f^(-1)(x)=(5(x-8))^(7) f^(-1)(x)=5(x-8)^(7)

Write function as y: To find the inverse function, we first write the function as y: y = (x^(1/7))/5 + 8Swap x and y: Next, we swap x and y to find the inverse: x = (y^(1/7))/5 + 8Solve for y: Now, we solve for y to get the inverse function. Start by subtracting 8 from both sides: x - 8 = (y^(1/7))/5Isolate seventh root: Multiply both sides by 5 to isolate the seventh root of y: 5(x - 8) = y^(1/7)Eliminate seventh root: Raise both sides to the power of 7 to eliminate the seventh root: [5(x - 8)]^7 = yFind inverse function: The inverse function is then: f^(-1)(x) = [5(x - 8)]^7

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Q. For the function

f(x)=\frac{\sqrt[7]{x}}{5}+8

, find

f^{-1}(x)

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f^{-1}(x)=5 x^{7}-8

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f^{-1}(x)=(5 x)^{7}-8

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f^{-1}(x)=(5(x-8))^{7}

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f^{-1}(x)=5(x-8)^{7}

Write function as $y$ : To find the inverse function, we first write the function as $y$ : $y = \frac{x^{\frac{1}{7}}}{5} + 8$
Swap x and y: Next, we swap x and y to find the inverse: $x = \frac{y^{\frac{1}{7}}}{5} + 8$
Solve for y: Now, we solve for y to get the inverse function. Start by subtracting $8$ from both sides: $x - 8 = \frac{y^{\frac{1}{7}}}{5}$
Isolate seventh root: Multiply both sides by $5$ to isolate the seventh root of $y$ : $5(x - 8) = y^{\frac{1}{7}}$
Eliminate seventh root: Raise both sides to the power of $7$ to eliminate the seventh root: $[5(x - 8)]^7 = y$
Find inverse function: The inverse function is then: $f^{-1}(x) = [5(x - 8)]^7$