Answer
The required proof is given below.
Work Step by Step
(a)
For this part we are given that the function $f(x)$ is linear with slope, $m$.
Thus, we can write $f(x)=m\cdot x +c$, where $c$ is the y-intercept of the function.
Now, for any $h$,
$f(x+h)=m\cdot (x+h)+c=m\cdot x +m\cdot h +c$ [using defined function]
$\Rightarrow f(x+h)=(m\cdot x +c)+m\cdot h$ [ by shifting terms]
$\Rightarrow f(x+h)=f(x)+m\cdot h$ [As, $f(x)=m\cdot x+c$]
$\Rightarrow f(x+h)-f(x)=m\cdot h$. This is valid for all $x$ and $h$.
Hence, proved.
(b)
Now, for all $x$ and $h$, we have $f(x+h)-f(x)=m\cdot h$.
Thus,
$\dfrac{f(x+h)-f(x)}{h}=m$
$\Rightarrow \dfrac{f(x+h)-f(x)}{(x+h)-x}=m$
$\Rightarrow \dfrac{\Delta f}{\Delta x}=m$, which is constant.
Since, the change in function with respect to change in $x$ is constant, which is only possible for a linear function. Therefore, $f(x)$ is a linear function.
