What is the inverse function of #y=3x^2-5#?
2 Answers
Explanation:
To find the inverse of a function expressed as
rearrange the expression into the form:
Given
Switch sides and add
Divide both sides by
Take the square root of both sides
Since we were asked for an inverse function, I've eliminated the non-primary root (a function can not have two values for a single argument value).
Therefore
With the default domain, this function has no inverse since it is not one-to-one, but read on...
Explanation:
Let
The implicit domain of
By convention, since the variable name is
However,
We can try to find an inverse as follows:
Let
Adding
#y+5 = 3x^2#
Dividing both sides by
#x^2 = (y+5)/3#
Hence
#x = +-sqrt((y+5)/3)#
This is not uniquely defined, so does not define a function, unless...
If we restrict the domain of
#f^(-1)(y) = sqrt((y+5)/3)#
Alternatively, if we restrict the domain of
#f^(-1)(y) = -sqrt((y+5)/3)#
Interestingly, if we define
If we define
If we define:
then
The implicit domain for