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Question
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
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Solution
\[f\left( x \right) = ax + b\]
\[\text { Let }x_1 , x_2 \text { in R such that } x_1 < x_2 . \]
\[\text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow a x_1 > a x_2 (\because a<0)\]
\[ \Rightarrow a x_1 + b > a x_2 + b\]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\text { Thus }, x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in R\]
\[\text { So },f\left( x \right) \text { is decreasing on R } .\]
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