Question

Graphs of modulus function f(x) =|x-2|-2

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is .

$\rm :\longmapsto\:f(x) = |x - 2| - 2$

$\rm :\longmapsto\:Let \: y \: = \: f(x) = |x - 2| - 2$

☆ Let first define the function f(x).

We know,

☆ By definition of Modulus function,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x| = \begin{cases} &\sf{x \: \: if \: x \geqslant 0} \\ &\sf{ - x \: \: if \: x < 0} \end{cases}\end{gathered}\end{gathered}$

Hence,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x - 2| = \begin{cases} &\sf{x - 2 \: \: if \: x - 2 \geqslant 0} \\ &\sf{ - (x - 2) \: \: if \: x - 2< 0} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x - 2| = \begin{cases} &\sf{x - 2 \: \: if \: x \geqslant 2} \\ &\sf{2 - x \: \: if \: x < 2} \end{cases}\end{gathered}\end{gathered}$

So,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x - 2| - 2= \begin{cases} &\sf{x - 2 - 2 \: \: if \: x \geqslant 2} \\ &\sf{2 - x - 2 \: \: if \: x < 2} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x - 2| - 2= \begin{cases} &\sf{x - 4 \: \: if \: x \geqslant 2} \\ &\sf{- x \: \: if \: x < 2} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\:\bf\implies \:y = |x - 2| - 2= \begin{cases} &\sf{x - 4 \: \: if \: x \geqslant 2} \\ &\sf{- x \: \: if \: x < 2} \end{cases}\end{gathered}\end{gathered}$

Case :- 1

$\rm :\longmapsto\:When \: x \geqslant 2$

$\rm :\longmapsto\:y = x - 4$

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 2 & \sf - 2 \\ \\ \sf 3 & \sf - 1 \\ \\ \sf 4 & \sf 0 \end{array}} \\ \end{gathered}$

☆ Blue line represents y = x - 4

Case :- 2

$\rm :\longmapsto\:When \: x \: < \: 2$

$\rm :\longmapsto\:y = - \: x$

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 1 & \sf - 1 \\ \\ \sf 0 & \sf 0 \\ \\ \sf - 1 & \sf 1 \end{array}} \\ \end{gathered}$

☆ Green line represents y = - x

➢ Now draw a graph using the points

➢ See the attachment graph.

Range

We know,

$\rm :\longmapsto\: |x - 2| \geqslant 0$

$\rm :\longmapsto\: |x - 2| - 2\geqslant 0 - 2$

$\rm :\longmapsto\: |x - 2| - 2\geqslant - 2$

$\bf\implies \:graph \: should \: lie \: above \: y \geqslant - 2$