Visual representation of the domain and range of a function?

Question

An excerpt in the book "College Algebra by Michael Sullivan is that:

When the graph of a function is given, its domain may be viewed as the shadow created by the graph on the x- axis by vertical beams of light. Its range can be viewed as the shadow created by the graph on the y-axis by horizontal beams of light.

I did't get it and i also sought in google to get some idea but i can't find one. Can anyone help me to understand it maybe with some graphics.

Thanks.

Answer 1

Consider a particular point on the function. If you cast a vertical light beam through that point it would cast a shadow on the x-axis according to the x value of that point. Since the domain is the set of all possible x values of a function, imagine repeating the vertical light beams on every point on the function. Then the resulting shadow on the x-axis represents the domain. It is a set of values on the x-axis. By a similar process you can represent the range on the y-axis.

Answer 2

The naive definition of domain and range are represented in the following figure

• the domain is the set of all $x$ values such that $y=f(x)$ is defined (and unique)

• the range is the set of all $y$ values such that $\exists x$ (at least one) and $y=f(x)$

Answer 3

This is not meant to be difficult. It says that going perpendicularly down/up from each $(x,f(x))$ (parallel to $y $-axis) we reach the $x$ point (point of which $f(x) $ is the image).

So 'projecting' all $(x,f(x))$ onto the $x $-axis gives the domain.

And similarly for the 'shadow' on the $y$-axis.