A function accepts values, performs particular operations on these values and generates an output. The inverse function accepts the resultant, performs an operation and reaches back to the original function.
If you consider f and g are inverse functions, f(g(x)) = g(f(x)) = x. A function that consists of its inverse functions fetches the original value.
How to find the inverse of a function?
Generally, the method of calculating an inverse is swapping of coordinates x and y. This newly created inverse is a relation but not necessarily a function.
The inverse of a function – The relation that is developed when the independent variable is interchanged with the variable that is dependent in a specified equation and this inverse may or may not be a function.
Inverse function – If the inverse of a function is a function by itself, then it is known as inverse function, denoted by f^{-1}(x).
The original function has to be a one-to-one function to assure that its inverse will be also a function. A function is said to be a one-to-one function only if every second element corresponds to first value. (values of x and y are used only once)
You can apply on the horizontal line test to verify whether a function is one-to-one function. If a horizontal line intersects the original function in a single region, the function is a one-to-one function and inverse is also a function.
Example:
To solve for an equation:
f(x) = 2x + 3, at x = 4
We have,
f(4) = 2 x 4 + 3
f(4) = 11
Now, lets apply for reverse on 11.
f^{-1}(11) = (11 – 3) / 2
f^{-1}(11) = 4
Magically we get 4 again
Therefore, f^{-1}(f(4)) = f(4)
So, when we apply function f and its reverse f^{-1} gives the original value back again, i.e, f^{-1}(f(x)) = x
An example illustrating inverse functions using Algebra.
Put “y” for “f(x)” and solve for x:
The function: | f(x) | = | 2x+3 | |
Put “y” for “f(x)”: | y | = | 2x+3 | |
Subtract 3 from both sides: | y-3 | = | 2x | |
Divide both sides by 2: | (y-3)/2 | = | x | |
Swap sides: | x | = | (y-3)/2 | |
Solution (put “f-1(y)” for “x”) : | f-1(y) | = | (y-3)/2 |
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