Question

What is the inverse of the function f(x) = ln (8x – 4) for {x: x ∈ R | x > 0.5}?
1. \({f^{ - 1}}\left( {\text{x}} \right) = \frac{{{e^x}}}{8}\)
2. \({f^{ - 1}}\left( {\text{x}} \right) = \frac{{{e^x} - 4}}{8}\)
3. \({f^{ - 1}}\left( {\text{x}} \right) = \frac{{{e^x} + 8}}{4}\)
4. \({f^{ - 1}}\left( {\text{x}} \right) = \frac{{{e^x} + 4}}{8}\)

Answer 1

Correct Answer - Option 4 : \({f^{ - 1}}\left( {\text{x}} \right) = \frac{{{e^x} + 4}}{8}\)

CONCEPT:

A function f: X → Y is defined to be invertible, if there exists a function g: Y → X such that g o f = I_x and

f o g = I_Y. The function g is called the inverse of f and is denoted by f ^–1.

• For calculation of inverse of any function, it should be arranged in the terms of x = f(y) and then every y should be replaced with x.

CALCULATIONS:

Given function is f(x) = ln (8x – 4) = y (say)

∴ y = ln (8x – 4)

e ^y = 8x – 4  ⇒ \(x = \;\frac{{{e^y} + 4}}{8} = {f^{ - 1}}\left( {\text{y}} \right)\)  

∴ \({f^{ - 1}}\left( {\text{x}} \right) = \frac{{{e^x} + 4}}{8}\)