### Home > CCA2 > Chapter 5 > Lesson 5.2.5 > Problem5-113

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.

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

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

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