4.3.2 Writing Statements from Function Notation

Extending Your Thinking

Interpreting Equations in Function Notation

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 started 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 $p(36)$ to mean “there are 36 blog subscribers after $x$ months,” while it actually means “after 36 months, there are $x$ thousands of blog subscribers.”