Using Function Notation II

No Tags

Alignments to Content Standards: F-IF.A.2

Task

Given a function $f$, is the statement $$f(x+h)=f(x)+f(h)$$ true for any two numbers $x$ and $h$? If so, prove it. If not, find a function for which the statement is true and a function for which the statement is false.

IM Commentary

The purpose of the task is to explicitly identify a common error made by many students, when they make use of the "identity" $f(x + h) = f(x) + f(h).$ A function $f$ cannot in general be distributed over a sum of inputs. This is an easy mistake to make because $$ f(x+h) = f(x) + f(h) $$ is a true statement if $f,x,h$ are real numbers and the operations implied by the parentheses are multiplication. The task has students find a single explicit example for which the identity is false, but it is worth emphasizing that in fact the identity fails for the vast majority of functions. Among continuous functions, the only functions satisfying the identity for all $x$ and $h$ are the functions $f(x)=ax$ for a constant $a$.

Solution

The statement is not true for all functions.

A function for which it holds is the function $f$ given by $f(a) = 5a$. If $f(a) = 5a$, then $$f(x+h) = 5(x+h) = 5 \cdot x + 5 \cdot h = f(x) + f(h).$$
A function for which the statement does not hold is the function $f$ given by $f(a) = a^2$. If $f(a) = a^2$, then $$f(x+h) = (x+h)^2=x^2 + 2xh + h^2.$$ This differs from $$f(x)+f(h)=x^2 + h^2$$ by $2xh$. This middle term is not zero unless $x$ or $h$ is zero.