- #1
hahaha158
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Homework Statement
f(x)=e^x^2
f^(-1)(x)=?
2. The attempt at a solution
i reversed x and y so i got x=e^y^2
ln both sides to get
lnx=y^2
so (ln(y))^(1/2)
what am i doing wrong?
well that last line does not follow from the one before it does it? You are trying to find y(x) remember.hahaha158 said:Homework Statement
##f(x)=e^{x^2}##
##f^{-1}(x)=?##2. The attempt at a solution
i reversed x and y so i got ##x=e^{y^2}##
ln both sides to get
##\ln|x|=y^2##
so ##[\ln|y|]^{1/2}##
what am i doing wrong?
The inverse of a function allows us to reverse the input and output values of a function, which can be useful for solving equations and understanding the behavior of the original function.
A function has an inverse if and only if it is one-to-one, meaning that each input value corresponds to only one output value. This can be determined by looking at the graph of the function or by using the horizontal line test.
No, not all functions have an inverse. Functions that are not one-to-one, such as quadratic functions and exponential functions, do not have an inverse. Additionally, some functions may have restricted domains that prevent them from having an inverse.
To find the formula for the inverse of a function, we can follow a simple process. First, replace the function notation with "y." Then, swap the positions of x and y. Finally, solve for y to get the inverse function in terms of x.
The inverse of a function is essentially the "mirror image" of the original function. This means that the inverse function will have the same shape and graph as the original, but it will be reflected over the line y = x. Additionally, the composition of a function and its inverse will always result in the input value (x) being returned, and vice versa.