Check whether f(x) is strictly increasing or decreasing.

 

(i) $f(x)=3x-2$ where $f:\mathbb{R}\to \mathbb{R}$

# Method 1:

Let $x_1,x_2$ be any element of $\mathbb{R}$ such that:

$x_1>x_2$

$3x_1>3x_2$

$3x_1-2>3x_2-2$

$f(x_1)>f(x_2)\mathrm{\forall }{x}_1,x_2\in \mathbb{R}$

So, $f(x)$ is strictly increasing on $\mathbb{R}.$

# Method 2:

$f(x)=3x-2$

$f^{\prime }(x)=3$

$$ As, $f^{\prime }(x)>0\mathrm{\forall }{x}_1,x_2\in \mathbb{R}$

So, $f(x)$ is strictly increasing on $\mathbb{R}.$

(ii) $f(x)=3-2x$ where $f:\mathbb{R}\to \mathbb{R}$

# Method 1:

Let $x_1,x_2$ be any element of $\mathbb{R}$ such that:

$x_1>x_2$

$-2x_1<-2x_2$

$3-2x_1<3-2x_2$

$f(x_1)<f(x_2)\mathrm{\forall }{x}_1,x_2\in \mathbb{R}$

So, $f(x)$ is strictly decreasing on $\mathbb{R}.$

# Method 2:

$f(x)=3-2x$

$f^{\prime }(x)=-2$

$$ As, $f^{\prime }(x)<0\mathrm{\forall }{x}_1,x_2\in \mathbb{R}$

So, $f(x)$ is strictly decreasing on $\mathbb{R}.$