Question

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Answer 1

Answer:

  A quadratic function generally does not have an inverse except on a restricted domain. If the domain of the original function is restricted to x ≥ 0, then the inverse function is ...

  $f^{-1}(x)=\displaystyle\sqrt{\frac{x-4}{5}}$

Step-by-step explanation:

Start by interchanging x and "y", then solve for y.

  y = 5x² +4 . . . . . . given

  x = 5y² +4 . . . . . . with x and y interchanged

  x -4 = 5y² . . . . . . subtract 4

  (x -4)/5 = y² . . . . . divide by 5

  √((x -4)/5) = y . . . take the square root . . . . x ≥ 4, y ≥ 0

Then, ...

  $f^{-1}(x)=\displaystyle\sqrt{\frac{x-4}{5}}$

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The graph shows the original f(x) and the "inverse", called f1(x). Note that it is only a reflection across y=x of the right half of the original function. That is, the inverse only exists for the original function when its domain is restricted to x ≥ 0. (The domain of the inverse function is x ≥ 4.)