Proof: $\sqrt{4} + \sqrt{9} = 2 + 3 = 5 \not= \sqrt{13} = \sqrt{4 + 9}$
Example: $h :R \rightarrow R$, $h(x) = \frac{1}{x}$ is NOT linear.
Proof: $ \frac{1}{1 + 2} = \frac{1}{3} \not= \frac{3}{2} = \frac{1}{1} + \frac{1}{2}$
Defn: A function $f$ is linear if
1.) $f(a{\bf x}) = af({\bf x})$ and
2.) $f({\bf x} + {\bf y}) =
f({\bf
x}) + f({\bf y})$
Example: $f :R \rightarrow R$, $f(x) = 2x$ is linear.
Proof:
1.) $f(a{ x}) = 2ax = a(2x) = af({ x})$ and
2.) $f(x + y) = 2(x + y) = 2x + 2y = f(x) + f(y)$
Example: $f :R \rightarrow R$, $f(x) = 2x + 3$ is NOT linear.
Proof: $f(2 \cdot 0) = f(0) = 3$, but $2f(0) = 2 \cdot 3 = 6$.
Hence $f(2 \cdot 0) \not= 2f(0)$
Alternate Proof: $f(0 + 1) = f(1) = 5$, but $f(0) + f(1) = 3 + 5 = 8$.
Hence $f(0 + 1) \not= f(0) + f(1)$
Note confusing notation: Most lines, $f(x) = mx + b$ are not linear
functions.
Question: When is a line, $f(x) = mx + b$, a linear function?
Note: To prove a function is linear, you need to show that
1.) $f(a{ x}) = af({ x})$ and
2.) $f(x + y) = f(x) + f(y)$
for all $x, y$ in domain of $f$.
Hence to prove that a function is not linear, you only need to find specific values where either (1) or (2) do not hold.