Question:
Prove that $f(x)=a x+b$, where $a$, $b$ are constants and $a>0$ is an increasing function on $R$.
Solution:
we have,
$f(x)=a x+b, a>0$
let $x_{1}, x_{2} \in R$ and $x_{1}>x_{2}$
$\Rightarrow \mathrm{ax}_{1}>\mathrm{ax}_{2}$ for some $\mathrm{a}>0$
$\Rightarrow \mathrm{ax}_{1}+\mathrm{b}>\mathrm{ax}_{2}+\mathrm{b}$ for some $\mathrm{b}$
$\Rightarrow \mathrm{f}\left(\mathrm{x}_{1}\right)>\mathrm{f}\left(\mathrm{x}_{2}\right)$
Hence, $x_{1}>x_{2} \Rightarrow f\left(x_{1}\right)>f\left(x_{2}\right)$
So, $f(x)$ is increasing function of $R$
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