Prove that f(x) = ax + b, where a, b are constants and a > 0 is a… - Vidyadip

Question

Prove that $f(x) = ax + b,$ where $a, b$ are constants and $a > 0$ is an increasing function on $R.$

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Answer

Here,
$f(x) = ax + b$
Let $\text{x}_1,\text{x}_2\in\text{R}$ such that $x_1< x_2.$ Then,
$x_1< x_2$
$\Rightarrow ax_1 < ax_{2 }[\because\ \text{a}>0]$
$\Rightarrow ax_1 + b < ax_2 + b$
$\Rightarrow f(x_1) < f(x_2)$
$\therefore x_1 < x_2$
$\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2),\forall\text{x}_1,\text{x}_2\in\text{R}$
So, $f(x)$ is increasing on $R.$

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