Computation of images and inverse images

Let's consider the function $f (x) = x^{2} + 1$ .

From its analytical expression we can compute the image of any element $x$ in the domain. To do so, it is enough to replace the value of $x$ in the expression of the function.

Example

For $x = 2$ : $f (2) = 2^{2} + 1 = 4 + 1 = 5$

Therefore, $5$ is the image of $2$ in $f$ .

We will write $f (2) = 5$ .

We can calculate also the inverse image or the images of any element $y$ of the codomain. To do so, it is enough to replace the value of $y = f (x)$ in the expression of the function and to solve $x$ .

Example

For example, the inverse image of $y = 10$ is: $\begin{array}{rcl} 10 & = & x^{2} + 1 \\ x^{2} & = & 9 \\ x & = & \pm 3 \end{array}$

Therefore, $3$ and $- 3$ are inverse images of $10$ for the function $f$ . We will write: $f^{- 1} (10) = {- 3, 3}$

Example

Compute the image of $2$ and the inverse image of $11$ for the function from the previous example $f (x) = 3 x^{2} - 1$ .

$f^{- 1} (11)$ : $\begin{array}{rcl} 11 & = & 3 x^{2} - 1 \\ 12 & = & 3 x^{2} \\ x^{2} & = & 4 \\ x & = & \pm 2 = {- 2, 2} \end{array} ⟹ f^{- 1} (11) = {- 2, 2}$