Question

Find the inverse of each function

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

2. f(x)= -2x^5 - 3

Answer 1

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Answer:

1. f^-1(x) = 4/(x+2) -2

2. f^-1(x) = (-(x+3)/2)^(1/5)

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]{(-(x+3))/(2)}\\\\\text{or}\\\\f^(-1)(x)=-(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.