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For the function 
f(x)=5x^(7)+9, find 
f^(-1)(x).

f^(-1)(x)=root(7)((x)/(5)-9)

f^(-1)(x)=root(7)((x-9)/(5))

f^(-1)(x)=(root(7)(x)-9)/(5)

f^(-1)(x)=(root(7)(x-9))/(5)

For the function f(x)=5x7+9 f(x)=5 x^{7}+9 , find f1(x) f^{-1}(x) .\newlinef1(x)=x597 f^{-1}(x)=\sqrt[7]{\frac{x}{5}-9} \newlinef1(x)=x957 f^{-1}(x)=\sqrt[7]{\frac{x-9}{5}} \newlinef1(x)=x795 f^{-1}(x)=\frac{\sqrt[7]{x}-9}{5} \newlinef1(x)=x975 f^{-1}(x)=\frac{\sqrt[7]{x-9}}{5}

Full solution

Q. For the function f(x)=5x7+9 f(x)=5 x^{7}+9 , find f1(x) f^{-1}(x) .\newlinef1(x)=x597 f^{-1}(x)=\sqrt[7]{\frac{x}{5}-9} \newlinef1(x)=x957 f^{-1}(x)=\sqrt[7]{\frac{x-9}{5}} \newlinef1(x)=x795 f^{-1}(x)=\frac{\sqrt[7]{x}-9}{5} \newlinef1(x)=x975 f^{-1}(x)=\frac{\sqrt[7]{x-9}}{5}
  1. Write Function: First, let's write down the function we're dealing with: f(x)=5x7+9f(x) = 5x^7 + 9. We need to find the inverse of this function, which we'll denote as f1(x)f^{-1}(x). To do this, we'll first replace f(x)f(x) with yy for simplicity: y=5x7+9y = 5x^7 + 9.
  2. Swap x and y: Now, we'll swap x and y to start finding the inverse: x=5y7+9x = 5y^7 + 9.
  3. Isolate yy: Next, we need to isolate yy on one side. So, we'll subtract 99 from both sides: x9=5y7x - 9 = 5y^7.
  4. Divide by 55: Then, we divide both sides by 55 to get: (x9)/5=y7(x - 9)/5 = y^7.
  5. Take 77th Root: Now, to solve for yy, we take the 77th root of both sides: y=x957y = \sqrt[7]{\frac{x - 9}{5}}.
  6. Replace yy: Finally, we replace yy with f1(x)f^{-1}(x) to denote the inverse function: f1(x)=x957f^{-1}(x) = \sqrt[7]{\frac{x - 9}{5}}.

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