# Using Function Notation II

## 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.

## Using Function Notation II

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.