Classify the following function as injection, surjection or bijection: \[\mathbf{f}:~\mathbf{Q}~-\text{ }\left\{ \mathbf{3} \right\}\text{ }\to ~\mathbf{Q}\], defined by \[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\left( \mathbf{2x}\text{ }+\mathbf{3} \right)/\left( \mathbf{x}-\mathbf{3} \right)\]

Classify the following function as injection, surjection or bijection:

$\mathbf{f}:~\mathbf{Q}~-\text{ }\left\{ \mathbf{3} \right\}\text{ }\to ~\mathbf{Q}$

, defined by

$\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\left( \mathbf{2x}\text{ }+\mathbf{3} \right)/\left( \mathbf{x}-\mathbf{3} \right)$

Classify the following function as injection, surjection or bijection:

$\mathbf{f}:~\mathbf{Q}~-\text{ }\left\{ \mathbf{3} \right\}\text{ }\to ~\mathbf{Q}$

, defined by

$\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\left( \mathbf{2x}\text{ }+\mathbf{3} \right)/\left( \mathbf{x}-\mathbf{3} \right)$

Given

$f:~Q~-\text{ }\left\{ 3 \right\}\text{ }\to ~Q$

, defined by

$f\text{ }\left( x \right)\text{ }=\text{ }\left( 2x\text{ }+3 \right)/\left( x-3 \right)$

Now we have to check for the given function is injection, surjection and bijection condition.

Injection test:

Let x and y be any two elements in the domain (Q − {

$3$

}), such that f(x) = f(y).

f(x) = f(y)

$\left( 2x\text{ }+\text{ }3 \right)/\left( x\text{ }\text{ }3 \right)\text{ }=\text{ }\left( 2y\text{ }+\text{ }3 \right)/\left( y\text{ }\text{ }3 \right)$

$\left( 2x\text{ }+\text{ }3 \right)~\left( y\text{ }-\text{ }3 \right)\text{ }=~\left( 2y\text{ }+\text{ }3 \right)~\left( x\text{ }-\text{ }3 \right)$