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Formulas for Inverse in Aptitude
Here , In this Page Formulas for Inverse is given.
Inverse is a Function that Reverses another Function.
If the function is denoted by f(x) then inverse is denoted by f^{-1}(x)
Inverse functions gives lots of troubles so here’s a swift run down of what an inverse function is, and how to find it.
- For a function to have an inverse, the function has to be 1 to 1. That means every output only has one input.
There are many Applications of Inverse Functions .It is used in Trigonometry for various Trigonometric Functions.


Inverse Formulas Example-
f(x) = 3x -2 and g(x) =\frac{x}{3} + \frac{2}{3}
(f∘g)(x)= (g∘f)(x) = x
- The first case is really,
(g∘f)(−1)=g[f(−1)]=g[−5]=−1
- The second case is really,
(f∘g)(2) = f(2) = f[g(2)] = f [ \frac{4}{3} \ ] = 2
- A function is called one-to-one if no two values of x produce the same y. Mathematically this is the same as saying
f(x1)\neq f(x2) whenever x1 \neq x2 - Inverse functions are one to one functions f(x) and g(x) if
(f∘g)(x) = x and (g∘f)(x) = x, then we can say f(x) and g(x) are inverse to each other. - g(x) is inverse of f(x) and denoted by
g(x)= f-1 (x) - Like wise f(x) is the inverse of g(x) are denoted by
f(x)= g-1 (x)
For the two functions that we started off this section with we could write either of the following two sets of notation.
f(x) = 3x -2
f-1 (x) = \frac{x}{3} + \frac{2}{3}
g(x) = \frac{x}{3} + \frac{2}{3}
g-1 (x) = 3x-2
Inverse Formulas with solved Examples
Given f(x) 3x – 2 find \mathbf{f^{-1} (x)}
Solution:
we’ll first replace f(x) with y
y= 3x-2
Next, replace x with y and all y with x.
x = 3y -2
x + 2 = 3y
\frac{1}{3} (x+2) = y
\frac{x}{3} + \frac{2}{3} = y
Finally, replace y with f^{-1} (x)
f^{-1} (x) = \frac{x}{3} \ + \frac{2}{3}
f∘f^{-1} (x) = f[f^{-1}(x)]
= f[ \frac{x}{3} \ + \frac{2}{3} \ ]
= 3 ( \frac{x}{3} \ + \frac{2}{3} \ ) – 2
= x + 2 – 2
= x
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