What to Do with Inverse Functions in Complex Cases?

In summary, the speaker is looking for the inverse function of a given hyperbolic tangent. They mention that the first case has two equal signs and the second one is a split function. They then ask for guidance on what to do in these cases.
  • #1
transgalactic
1,395
0
i need to find the inverse function of these:
http://img522.imageshack.us/my.php?image=63348338nf8.gif

i know that if y=sinx
then its inverse whould be x=arcsin y

but the first case has two = signs
and the second one is a split function

what to do in these cases?
 
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  • #2
In the first case, you are being given the definition of the hyperbolic tangent. So one answer is x = arctanh y. But presumably you are to ignore this, and solve it as if you were given only the function on the far right.

As for the split function, for each specified range of x, what is the corresponding range of y?
 

FAQ: What to Do with Inverse Functions in Complex Cases?

What is an inverse function?

An inverse function is a function that "undoes" the action of another function. In other words, if a function f(x) maps input values to output values, then its inverse function, denoted as f^-1(x), maps those output values back to the original input values.

How do I find the inverse of a function?

To find the inverse of a function, follow these steps: 1) Replace f(x) with y, 2) Swap the x and y variables, 3) Solve for y, 4) Replace y with f^-1(x). The resulting function is the inverse of the original function.

What is the domain and range of an inverse function?

The domain of an inverse function is the same as the range of the original function, and the range of an inverse function is the same as the domain of the original function. In other words, the input and output values are swapped between the original and inverse functions.

Can all functions have an inverse?

No, not all functions have an inverse. For a function to have an inverse, each input value must have a unique output value. This means that the function must pass the horizontal line test, where no horizontal line can intersect the graph of the function more than once.

How do inverse functions relate to composition of functions?

The composition of two functions, denoted as f(g(x)), is equivalent to applying the first function (g(x)) and then the second function (f(x)). When finding the inverse of a composition of functions, the order of the functions is reversed, resulting in f^-1(g^-1(x)).

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