Question

Find the inverse of each function 1. f(x) = 4/(x+2) - 2 2. f(x)= -2x^5 - 3

Answer 1

We are given the following function:

f(x) = 4/(x+2) - 2

We are asked to determine the inverse of this function. To do that we will first change the "f(x)" for "y":

$y = \frac{4}{x + 2} - 2$

Now we will switch "x" and "y", like this:

$x = \frac{4}{x + 2} - 2$

Now we will solve for "y", first by adding 2 to both sides:

$x + 2 = \frac{4}{y + 2}$

Now we multiply both sides by "y+2":

$(y + 2)(x + 2) = 4$

Now we divide both sides by "x+2":

$y + 2 = \frac{4}{x + 2}$

Now we subtract 2 to both sides:

$y = \frac{4}{x + 2} - 2$

Now we change "y" for the inverse of f(x), that is:

${f}^{ - 1} (x) = \frac{4}{x + 2} - 2$

And thus we found the inverse. A similar procedure can be used for function 2.

Answer 2

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Answer:

  1. f^-1(x) = 4/(x+2) -2

  2. f^-1(x) = (-(x+3)/2)^(1/5)

Step-by-step explanation:

1. As with all "inverse function" problems, solve for y:

  x = f(y)

  x +2 = 4/(y +2) . . . . add 2

  y +2 = 4/(x +2) . . . . . multiply by (y+2)/(x+2)

  y = 4/(x+2) -2 . . . . . subtract 2

We see that this function is its own inverse. The attached graph shows it is symmetrical about the line y=x.

  f^-1(x) = 4/(x+2) -2

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2. x = f(y)

  x +3 = -2y^5 . . . . add 3

  -(x +3)/2 = y^5 . . . . . divide by 2

  (-(x +3)/2)^(1/5) = y . . . . take the 5th root

  f^-1(x) = (-(x +3)/2)^(1/5)

In typeset form, that is ...

  $\displaystyle f^{-1}(x)=\sqrt[5]{\frac{-(x+3)}{2}}\\\\\text{or}\\\\f^{-1}(x)=-\frac{1}{2}\sqrt[5]{16x +48}$

This last version is with the denominator "rationalized" and the contents of the radical "simplified." It may be a preferred form.

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The graphs show the function and inverse are symmetrical about the line y=x, as they should be.