Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing 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 decreasing function of R

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