Finding the Inverse of a Function with a Trigonometric Term

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In summary, the conversation is about finding the inverse of a function that includes a cosine term. The person asking for help is struggling to solve the original equation for x and is unsure how to get x by itself when it is trapped within the cosine function. They mention that they used to know how to do this, but have forgotten. Someone else responds that it may not be possible to find a nice expression for the inverse function and suggests answering questions about the inverse without having an expression for it. The person asking for help then mentions that the question in their book is asking to find f^-1(1) and wonders if they should just plug in 1 for y and solve. They also mention that this would likely only leave 0 as
  • #1
Joe_K
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Homework Statement



f(x)= x+cosx

find the inverse, f^-1(x)



The Attempt at a Solution



To start, I tried to solve the original equation for x. But this is where I am having trouble. How do you get x by itself when it is trapped within the cos function? I used to know how to do this but I seem to have forgotten. Once I solve the equation for x, I should be able to switch the "x" and "y" terms and be left with the inverse of the original function. Maybe someone can help me remember how to get the 'x' by itself. Thank you.
 
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  • #2
Your memory of how to do it is fine, but you can't solve this one for x algebraically.
 
  • #3
What do I need to do, in order to do the problem correctly? The question in my book is asking to find f^-1(1). Should I just plug in '1' for y in the equation and solve? I believe that would only leave zero as a possible answer?
 
  • #4
Joe_K said:
What do I need to do, in order to do the problem correctly? The question in my book is asking to find f^-1(1). Should I just plug in '1' for y in the equation and solve? I believe that would only leave zero as a possible answer?

Undoubtedly the reason they asked that simpler question is because you can't do the general one. So, yes. Obviously x = 0 is a value solving 1 = x + cos(x). Do you see how to show it is the only solution or, for that matter, that the inverse function exists?
 
  • #5
Yes, thank you for your help!
 
  • #6
Joe_K said:

Homework Statement



f(x)= x+cosx

find the inverse, f^-1(x)



The Attempt at a Solution



To start, I tried to solve the original equation for x. But this is where I am having trouble. How do you get x by itself when it is trapped within the cos function? I used to know how to do this but I seem to have forgotten. Once I solve the equation for x, I should be able to switch the "x" and "y" terms and be left with the inverse of the original function. Maybe someone can help me remember how to get the 'x' by itself. Thank you.

I don't think you are going to find a nice expression for f^(-1)(x). You can probably answer some questions about f^(-1) without having an expression for it. Is that the whole question?
 

FAQ: Finding the Inverse of a Function with a Trigonometric Term

What is an inverse and why is it important?

An inverse is a mathematical operation that undoes another operation. It is important because it allows us to solve equations, find missing values, and perform other mathematical tasks.

How do I find the inverse of a function?

To find the inverse of a function, you can follow these steps: 1) Replace f(x) with y. 2) Switch the x and y variables. 3) Solve for y. 4) Replace y with f^-1(x). This will give you the inverse function.

Can every function have an inverse?

No, not every function has 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. If a function has repeating outputs for different inputs, it does not have an inverse.

How can I check if my inverse function is correct?

You can check if your inverse function is correct by plugging in values for x and y and seeing if they satisfy the original function and its inverse. For example, if your original function is f(x) = 2x+3 and your inverse function is f^-1(x) = (x-3)/2, plugging in x=5 for both functions should give you the same value for y.

Can I use the inverse of a function to solve equations?

Yes, the inverse of a function can be used to solve equations. By finding the inverse of a function, you can isolate the variable and solve for it. This is especially useful for solving equations involving trigonometric functions and logarithms.

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