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

For the function f(x)=72x+7 f(x)=\frac{7}{2 x+7} , find f−1(x) f^{-1}(x) .\newlineAnswer: f−1(x)= f^{-1}(x)=

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Q. For the function f(x)=72x+7 f(x)=\frac{7}{2 x+7} , find f−1(x) f^{-1}(x) .\newlineAnswer: f−1(x)= f^{-1}(x)=
  1. Rewrite function with y: To find the inverse function, f−1(x)f^{-1}(x), we need to switch the roles of xx and yy in the original function and then solve for yy. Let's start by rewriting the function with yy instead of f(x)f(x):\newliney=72x+7y = \frac{7}{2x + 7}
  2. Switch x and y: Now, switch x and y to find the inverse: x=72y+7x = \frac{7}{2y + 7}
  3. Multiply both sides: Next, we solve for yy. Start by multiplying both sides of the equation by (2y+7)(2y + 7) to get rid of the fraction:\newlinex(2y+7)=7x(2y + 7) = 7
  4. Distribute xx: Distribute xx on the left side of the equation: 2xy+7x=72xy + 7x = 7
  5. Isolate term with y: Now, we want to isolate the term with yy in it, so subtract 7x7x from both sides: 2xy=7−7x2xy = 7 - 7x
  6. Divide both sides: To solve for yy, divide both sides by 2x2x:y=7−7x2xy = \frac{7 - 7x}{2x}
  7. Simplify the expression: Finally, we can simplify the expression by factoring out the 77 in the numerator: y=7(1−x)2xy = \frac{7(1 - x)}{2x}
  8. Inverse function: This gives us the inverse function: f−1(x)=7(1−x)2xf^{-1}(x) = \frac{7(1 - x)}{2x}

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