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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 $h$ ____________"?

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    $\begingroup$ I would probably say "exponentiates its input with base $2$". $\endgroup$
    – Dave L Renfro
    Commented Oct 31, 2020 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 $2$". $\endgroup$
    – Xander Henderson ♦
    Commented Nov 1, 2020 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
    Commented Nov 1, 2020 at 18:46
  • $\begingroup$ "First, the input is divided by four. Then two is raised to the power of the result." $\endgroup$
    – Xander Henderson ♦
    Commented Nov 1, 2020 at 21:21

2 Answers 2

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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.

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"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.

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