Calculus

Let f(x)=(x/x+3).
Find Domain
Find f^(-1) and its domain
Verify f*f^(-1)=f^(-1)*f=x
Find (f(x+h)-f(x))/h.

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1. I bet you mean
y = x/(x+3)

The denominator is 0 when x = -3 so x = -3 is not in our domain. Otherwise the domain is all real numbers.

Inverse
x = y/(y+3)
x y + 3 x = y
y(x-1) = -3 x
y = -3x/(x-1)
domain is all real numbers except x = 1
f[f^(-1)] = f[-3x/(x-1) ]

= [-3x/(x-1) ] /{ [-3x/(x-1) ]+3 }

[-3x/(x-1) ]/{[-3x/(x-1) ]+3(x-1)/(x-1)}

= -3x/{-3x +3x -3}

= -3x/-3
= x
etc

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posted by Damon

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