# Clearest verb phrases for operations

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 $$h$$ ____________"?

• I would probably say "exponentiates its input with base $2$". Oct 31, 2020 at 15:45
• 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 $2$". Nov 1, 2020 at 4:20
• @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... Nov 1, 2020 at 18:46
• "First, the input is divided by four. Then two is raised to the power of the result." Nov 1, 2020 at 21:21

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