  ### Home > A2C > Chapter 6 > Lesson 6.2.5 > Problem6-114

6-114.

For functions of the form $f \left(x\right) = mx$, it is true that $f\left(a\right) + f\left(b\right) = f\left(a + b\right)$? For example, when $f\left(x\right) = 5x$, $f\left(a\right) + f\left(b\right) = 5a + 5b = 5\left(a + b\right)$ and $f\left(a + b\right) = 5\left(a + b\right)$. Is $f\left(a\right) + f\left(b\right) = f\left(a + b\right)$ 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 $f\left(a\right) + f\left(b\right) = f\left(a + b\right)$.

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

$f\left(a\right) + f\left(b\right) = \left(2a + 3\right) + \left(2b + 3\right)\\f(a + b) = 2(a + b) + 3\\f(a) + f(b) = 2a + 2b + 6$

But, $2a + 2b + 6 ≠ 2a + 2b + 3$

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