Describe in words what a variable is, and what a function is.

Triple the number, then subtract six.

Return 4, no matter what.

Add one. Then take the square root of the result. Then, divide that result into two.

Add two to the original number. Subtract two from the original number. Then, multiply those two answers together.

Subtract two, then triple.

Square, then subtract four.

Add three. Then, multiply by four. Then, subtract twelve. Then, divide by the original number.

In some of the above cases, two functions always give the same answer, even though they are different functions. We say that these functions are “equal” to each other. For instance, the function “add three and then subtract five” is equal to the function “subtract two” because they always give the same answer. (Try it, if you don’t believe me!) We can write this as:

x + 3 − 5 = x − 2 x + 3 − 5 = x − 2 size 12{x+3 - 5=x - 2} {}

Note that this is not an equation you can solve for x x —it is a generalization which is true for all x x values. It is a way of indicating that if you do the calculation on the left, and the calculation on the right, they will always give you the same answer.

In the functions #2-8 above, there are three such pairs of “equal” functions. Which ones are they? Write the algebraic equations that state their equalities (like my x+3−5=x−2 x 3 5 x 2 equation).

Of the following sets of numbers, there is one that could not possibly have been generated by any function whatsoever. Which set it is, and why? (No credit unless you explain why!)