Why Can We Swap Variables When Finding Inverse Functions?

This is related to the first thing you did because it is a way to find the inverse function by swapping the variables and solving for the new variable. In summary, we can find the inverse function of y= f(x) by swapping the variables and solving for the new variable to get y= g(x).
  • #1
hyper
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We have y=f(x), and get the inverse by uing the first function and solving it for x and get x=g(y). (F and g are different functions.) Then we swap the name of x and y and we get y=g(x).

Buw why can we do this when we want to find the inverse functions? If we got y=f(x) and want to find the inverse we take g(f(x))=x. But how is this related to the first thing I did?
 
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  • #2
Since y= f(x), g(f(x))= x is the same as g(y)= x.
 

FAQ: Why Can We Swap Variables When Finding Inverse Functions?

What is an inverse function?

An inverse function is a mathematical relationship between two variables, where the output of one function becomes the input of the other function, and vice versa. In simpler terms, it is a function that "undoes" another function.

How do you find the inverse of a function?

To find the inverse of a function, you can use the following steps:

  1. Replace the function notation with y.
  2. Switch the x and y variables.
  3. Solve for y.
  4. Replace y with the inverse function notation, f-1(x).

What is the domain and range of an inverse function?

The domain of an inverse function is the range of the original function, and the range of an inverse function is the domain of the original function. In other words, the inputs and outputs of an inverse function are switched compared to the original function.

What is the relationship between an inverse function and its original function?

An inverse function and its original function are reflections of each other over the line y=x. This means that if you graph both functions, they will intersect at this line and their outputs will be switched.

When can a function have an inverse?

A function can have an inverse if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once. This ensures that each input has only one output, which is necessary for a function to have an inverse.

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