# Formulas For Inverse

## Formulas for Inverse in Aptitude

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,

(gf)(1)=g[f(1)]=g[5]=1

• The second case is really,

(f

• 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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