CPM Homework Help : CCA2 Problem 5-113

For functions of the form $f(x)=mx$ , it is true that $f(a)+f(b)=f(a+b)$ . For example, when $f(x)=5x$ , $f(a)+f(b)=5a+5b=5(a+b)$ and $f(a+b)=5(a+b)$ . Is $f(a)+f(b)=f(a+b)$ true for all linear functions? Explain why or show why not.

Hint:

Think of a linear function that has a y-intercept other than $0$ and use it to check the relationship $f(a)+f(b)=f(a+b)$ .

More Help:

For example, try the linear function $f(x)=2x+3$ (you should think of a different one for your own answer).

$f(a)+f(b)=(2a+3)+(2b+3)$
$f(a+b)=2(a+b)+3$
$f(a)+f(b)=2a+2b+6$

But, $2a+2b+6\ne2a+2b+3$

Answer:

$f(a)+f(b)=f(a+b)$ is not true for all linear functions.