Inverse Functions: Reflection of f(x) & g(x) Logic

In summary, inverse functions are functions that perform the opposite operation of another function and are denoted by f^-1(x). To find the inverse of a function, you must replace f(x) with y, swap the x and y variables, solve for y, and then replace y with f^-1(x). Inverse functions are related to reflection as they are a mirror image of the original function over the line y=x. Not all functions have an inverse, only those with a one-to-one relationship between inputs and outputs. Inverse functions have practical uses in finance, engineering, physics, and cryptography.
  • #1
pan90
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Can anyone explain the logic behind the answer?

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  • #2
Suppose we have the point:

\(\displaystyle (x,f(x))\)

on the plot of \(f(x)\). Then, on the plot of \(g(x)\), we must have the corresponding point:

\(\displaystyle (f(x),x)\)

Now, consider that for all possible points, the locus of the mid-points is:

\(\displaystyle \left(\frac{x+f(x)}{2},\frac{x+f(x)}{2}\right)\)

Thereby implying that the line of symmetry must be:

\(\displaystyle y=x\)
 

What is the definition of an inverse function?

An inverse function is a function that undoes the action of another function. In other words, if a function f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes the output y and produces the original input x.

How do you find the inverse of a function?

To find the inverse of a function, you need to switch the input and output variables and solve for the new output. This can be done algebraically by swapping the x and y variables and solving for y, or graphically by reflecting the function over the line y=x.

What is the relationship between a function and its inverse?

The relationship between a function and its inverse is that they "undo" each other. This means that the composition of a function and its inverse will result in the original input. In other words, f(f^-1(x)) = x and f^-1(f(x)) = x.

What is the purpose of reflecting a function over the line y=x?

Reflecting a function over the line y=x allows you to find its inverse function. This is because the line y=x represents the "mirror" between a function and its inverse, and reflecting over this line essentially swaps the input and output variables.

Can every function have an inverse?

No, not every function has an inverse. A function must be one-to-one (each input has a unique output) in order to have an inverse. Functions that fail the horizontal line test (where a horizontal line intersects the function more than once) do not have an inverse.

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