Formulas For Inverse

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