3
$\begingroup$

What is the clearest way to indicate various operations along the following lines:

$f(x) = 3x$: The function $f$ multiplies its input by 3.

$g(x) = x-5$: The function $g$ decreases its input by 5.

$h(x) = 2^x$: The function $h$...Raises 2 to the power of its input? Exponentiates its input on a base of 2? Takes 2 to its input?

In the first two cases, I made a choice of whether to refer to the operation's name ("multiply") or a (hopefully) plain language action being performed ("decrease"). For the function $f$, I could have said that it triples its input or increases its input by 200%, but these don't seem to generalize well for the purposes of communicating, and the latter is almost never obvious to students. Similarly, I could have said that the function $g$ subtracts 5 from its input, but I am not convinced this is any simpler than "decreases its input by".

Is there a simplest verb phrase for exponents? What do you think is clearest for students if I want to maintain the form "The function $h$ ____________"?

$\endgroup$
4
  • 4
    $\begingroup$ I would probably say "exponentiates its input with base $2$". $\endgroup$ Oct 31 '20 at 15:45
  • $\begingroup$ Personally, I find all of the alternatives to be awkward, and would avoid verbing the phrase. I think it is clearer to say "$f$ is the exponential function with base $2$". $\endgroup$ Nov 1 '20 at 4:20
  • $\begingroup$ @XanderHenderson My motivation is when trying to explain an order of operations for something like $f(x)=3(2)^{x/4}+5$. First the input is divided by 4, then... $\endgroup$
    – Nick C
    Nov 1 '20 at 18:46
  • $\begingroup$ "First, the input is divided by four. Then two is raised to the power of the result." $\endgroup$ Nov 1 '20 at 21:21
4
$\begingroup$

If the domain was the real numbers, I would say that it raises two to the power of its input. If the domain was positive integers, I would break your model and say that it multiplied together $x$ copies of $2$ because I think it is more intuitive to describe what exponentiation actually does that to imply that it is as natural as addition and multiplication.

$\endgroup$
2
$\begingroup$

"The function $h$ raises $2$ to the $x$, where $x$ is the input"

and

"The function $h$ raises $2$ to the $x$th power, where $x$ is the input"

both ape the structure of $f$ and $g$ and, despite containing a clause, sound less awkward/contrived than the alternatives.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.