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

we have,

f(x) = ax + b, a > 0

let x_{1},x_{2} R and x_{1} > x_{2}

⇒ ax_{1} > ax_{2} for some a > 0

⇒ ax_{1} + b> ax_{2} + b for some b

⇒ f(x_{1}) > f(x_{2})

Hence, x_{1} > x_{2}⇒ f(x_{1}) > f(x_{2})

So, f(x) is increasing function of R

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