What does a statement like mean?

On its own, only tells us that when takes 3 as its input, its output is 12.

If we know what quantities the input and output represent, however, we can learn much more about the situation that the function represents.

• If function gives the perimeter of a square whose side length is and both measurements are in inches, then we can interpret to mean “a square whose side length is 3 inches has a perimeter of 12 inches.”

  We can also interpret statements like to mean “a square with side length has a perimeter of 32 inches,” which then allows us to reason that must be 8 inches and to write .

• If function gives the number of blog subscribers, in thousands, months after a blogger started publishing online, then means “3 months after a blogger starts publishing online, the blog has 12,000 subscribers.”

It is important to pay attention to the units of measurement when analyzing a function. Otherwise, we might mistake what is happening in the situation. If we miss that is measured in thousands, we might misinterpret to mean “there are 36 blog subscribers after months,” while it actually means “there are 36,000 subscribers after months.”

A graph of a function can likewise help us interpret statements in function notation.

Each point on the graph has the coordinates , where the first value is the input of the function and the second value is the output.

• represents the depth of water 2 minutes after the tub started being drained. The graph passes through , so the depth of water is 5 inches when . The equation captures this information.

• gives the depth of the water when the draining began, when . The graph shows the depth of water to be 6 inches at that time, so we can write .

• tells us that minutes after the tub started draining, the depth of the water is 3 inches. The graph shows that this happens when  is 6.