What does a statement like $p(3)=12$ mean?

On its own, $p(3)=12$ only tells us that when $p$ 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 $p$ gives the perimeter of a square whose side length is $x$ and both measurements are in inches, then we can interpret $p(3)=12$ to mean “a square whose side length is 3 inches has a perimeter of 12 inches.”

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

• If function $p$ gives the number of blog subscribers, in thousands, $x$ months after a blogger started publishing online, then $p(3)=12$ 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 $p(x)$ is measured in thousands, we might misinterpret $p(x)=36$ to mean “there are 36 blog subscribers after $x$ months,” while it actually means “there are 36,000 subscribers after $x$ months.”

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

Each point on the graph has the coordinates $(t, f(t))$, where the first value is the input of the function and the second value is the output.

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

• $f(0)$ gives the depth of the water when the draining began, when $t=0$. The graph shows the depth of water to be 6 inches at that time, so we can write $f(0)=6$.

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