Question

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

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ f\left(x\right)=ax+b \\ \text{Let}{x}_1,x_2\in R\text{such that}{x}_1<x_2.\text{Then,} \\ {x}_1<x_2 \\ \Rightarrow a{x}_1<a{x}_2\left[\text{∵}\mathit{\text{a}}\text{>0}\right] \\ \Rightarrow a{x}_1+b<a{x}_2+b \\ \Rightarrow f\left(x_1\right)<f\left(x_2\right) \\ \therefore x_1<x_2 \\ \Rightarrow f\left(x_1\right)<f\left(x_2\right),\forall x_1,x_2\in R \\ \text{So,}f\left(x\right)\text{is increasing on}R. \end{gathered}$$