# How to find the Inverse of f(x) = 3+x+(e^x)

• r_swayze
In summary, the conversation discusses finding the inverse of a function and how to solve for specific points on that inverse. The main difficulty is figuring out what the inverse represents and how to extract the desired value without knowing the explicit formula. It is suggested to use numerical methods to approximate the value in more complex cases.
r_swayze
?

Is it even possible?

Not explicitly, is that the full problem?

Well the full problem is:

If f(x) = 3+x+(e^x) , find f^-1(4)

so wouldn't I need to find the inverse first and then plug in 4?

No think about what f^(-1)(4) is, it's basically what value of x will give you 4 i.e. solve 3 + x + e^x = 4... there's an obvious value for x.

I know the answer is 0 if you just use trial and error or graph it out, but what if the problem was more complex? What I am having trouble with is how do solve for x when you have:

y = x + e^x

What would be the next step? I can't think of any way to extract the x

There isn't. That's the whole point of these problems, for you think what f^(-1)(blah) means. In this case it means that blah must be in the range, so they are trying to get you to figure out what it would map to in the domain without knowing explicitly what the formula would be. You COULD have approximated the value using any number of techniques if it wasn't something "nice"

You can't in any reasonably simple way. There aren't any standard functions in the book to write the answer with. If you want to find say f^(-1)(3) you just have to use numerical methods to get an approximation.

swayze, perhaps it'd be useful to think more simply, ie, what is the relationship of the points between two inverse functions? The question only asks about one particular point, so it's pointless to find out what the entire inverse function is. (Assuming that, since the question asks about an inverse, the inverse exists.)

## 1. What is the definition of the inverse of a function?

The inverse of a function is a new function that "undoes" the original function. It essentially switches the input and output values, so that the input of the original function becomes the output of the inverse function, and vice versa.

## 2. How do I find the inverse of a given function?

To find the inverse of a function, you can follow the steps of "switching" the input and output values. For example, in the function f(x) = 3+x+(e^x), you can rewrite it as y = 3+x+(e^x) and then solve for x in terms of y. This new equation will be the inverse function, denoted as f^-1(y).

## 3. Can any function have an inverse?

No, not all functions have an inverse. For a function to have an inverse, it must be a one-to-one function, meaning that each input has a unique output and vice versa. In other words, no two different inputs can have the same output. If a function is not one-to-one, it does not have an inverse.

## 4. How do I determine if a function is one-to-one and has an inverse?

To determine if a function is one-to-one, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one and does not have an inverse. If a horizontal line only intersects the graph at one point, then the function is one-to-one and has an inverse.

## 5. Is there a shortcut or easier way to find the inverse of a function?

Yes, you can use the inverse function rule, which states that if a function f(x) can be written as y = mx + b, then its inverse function is f^-1(y) = (x-b)/m. This rule can be applied to functions of the form f(x) = a+bx+cx^2+...+zx^n. However, it may not work for more complicated functions, so it is always recommended to follow the steps of "switching" the input and output values to find the inverse function.

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