Image of a function

Example

Consider the function $f(x)={\displaystyle \sqrt{x-3}}$ where we take the positive solution of the square root, only has positive images.

We will only consider those $x$ in the domain that belong to the real number, in which case all the real numbers that are greater than or equal to $0$.

Example

Therefore the image of the function $f(x)=\sqrt{x-3}$ is $Im(f)=[0,+\mathrm{\infty })$

Example

Find the image of the following functions:

1. $f(x)=2x-1$
2. $f(x)=3x^2$

3. $f(x)={\displaystyle \frac{1}{x}}$

4. The function can take any real number as an image. Therefore, $Im(f)={\displaystyle \mathbb{R}}$.

5. In this case the function only has positive images or $0$, since the square of a number cannot be a negative. Therefore $Im(f)=[0,+\mathrm{\infty })$

6. Finally, the function can take any real value except $0$, since it $f(x)={\displaystyle \frac{1}{x}}$ only when $x=0$ we have that the image is not defined in $\mathbb{R}$.

Therefore, $Im(f)=\mathbb{R}-0$.