Question

Prove that
$f(x) = log_{2}(3x + 2)$
is one to one function​

Answer 1

A function $f:A\to B$ is said to be a one - one function if and only if distinct elements of $A$ have distinct images in $B,$ i.e., $a_1\neq a_2\quad\!\iff\quad\!f(a_1)\neq f(a_2)\quad\forall a_1,\ a_2\in A$

Given,

$\longrightarrow f(x)=\log_2(3x+2)$

We need to check if it's a one - one function.

Assume,

$\longrightarrow f(x_1)\neq f(x_2)$

$\longrightarrow\log_2(3x_1+2)\neq\log_2(3x_2+2)$

Taking antilog,

$\longrightarrow3x_1+2\neq3x_2+2$

$\longrightarrow3x_1\neq3x_2$

$\longrightarrow x_1\neq x_2$

That is,

$\longrightarrow f(x_1)\neq f(x_2)\quad\!\implies\quad\!x_1\neq x_2$

On doing the reverse we can prove,

$\longrightarrow x_1\neq x_2\quad\!\implies\quad\!f(x_1)\neq f(x_2)$

Finally we get,

$\longrightarrow x_1\neq x_2\quad\!\iff\quad\!f(x_1)\neq f(x_2)$

Hence $f$ is a one - one function. Thus proved.