Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decr

1.	Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R.
Answer» Given as, f (x) = ax + b, a < 0 Suppose x₁, x₂ ∈ R and x₁ > x₂ ⇒ ax₁ < ax₂ for some a > 0 ⇒ ax₁ + b < ax₂ + b for some b ⇒ f (x₁) < f(x₂) Thus, x₁ > x₂⇒ f(x₁) < f(x₂) Therefore, f(x) is decreasing function of R

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

Answer»

Given as,

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

Suppose x₁, x₂ ∈ R and x₁ > x₂

⇒ ax₁ < ax₂ for some a > 0

⇒ ax₁ + b < ax₂ + b for some b

⇒ f (x₁) < f(x₂)

Thus, x₁ > x₂⇒ f(x₁) < f(x₂)

Therefore, f(x) is decreasing function of R