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Dovator [93]
3 years ago
14

Given f(x) = (-1/7)(sqrt 16-x^2), find f^-1(x). Then state whether the inverse is a function.

Mathematics
1 answer:
Viefleur [7K]3 years ago
8 0

Answer:

  • f^-1(x) = ±√(16 -49x^2) . . . . -4/7 ≤ x ≤  0
  • not a function

Step-by-step explanation:

The usual method of finding the inverse of a function is to solve for y the equation ...

  x = f(y)

The inverse is only defined on the range of the original function, so for ...

  0 ≥ x ≥ -4/7

Here, that looks like ...

  x = (-1/7)√(16 -y^2)

  -7x = √(16 -y^2) . . . multiply by -7

  49x^2 = 16 -y^2 . . . square both sides

  y^2 + 49x^2 = 16 . . . add y^2

  y^2 = 16 -49x^2 . . . . subtract 49x^2

  y = ±√(16 -49x^2) . . . . take the square root

So, the inverse function is ...

  f^-1(x) = ±√(16 -49x^2) . . . . . defined only for -4/7 ≤ x ≤ 0

This gives two output values for each input value, so is not a function.

_____

The original f(x) is the bottom half of an ellipse, so does not pass the horizontal line test. It cannot have an inverse function.

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