Let $f\left(x\right)=\{\begin{array}{cc}\hfill 2+2x,\hfill & \hfill x\in \left(-1,0\right)\hfill \\ \hfill 1-\frac{x}{3},\hfill & \hfill x\in [0,3)\hfill \end{array}$

$g\left(x\right)=\{\begin{array}{cc}\hfill x,\hfill & \hfill x\in [0,1)\hfill \\ \hfill -x,\hfill & \hfill x\in \left(-3,0\right)\hfill \end{array}$           

The range of fog(x) is

Correct Option - 3


Detailed Solution:

$f\left(x\right)=\{\begin{array}{cc}\hfill 2+2x,\hfill & \hfill x\in \left(-1,0\right)\hfill \\ \hfill 1-\frac{x}{3},\hfill & \hfill x\in [0,3)\hfill \end{array}$

$g\left(x\right)=\{\begin{array}{cc}\hfill x,\hfill & \hfill x\in [0,1)\hfill \\ \hfill -x,\hfill & \hfill x\in \left(-3,0\right)\hfill \end{array}$   ->g(x) = |x|, x Î (–3, 1)

$f\left(g\left(x\right)\right)=\{\begin{array}{cc}\hfill 2+2\left|x\right|,\hfill & \hfill \left|x\right|\in \left(-1,0\right)\Rightarrow x\in ?\hfill \\ \hfill 1-\frac{\left|x\right|}{3},\hfill & \hfill \left|x\right|\in [0,3)\Rightarrow x\in \left(-3,1\right)\hfill \end{array}$            

$f\left(g\left(x\right)\right)=\{\begin{array}{cc}\hfill 1-\frac{x}{3},\hfill & \hfill x\in [0,1)\hfill \\ \hfill 1+\frac{x}{3},\hfill & \hfill x\in \left(-3,0\right)\hfill \end{array}$

Range of fog(x) is [0, 1]

            

            Range of fog(x) is [0, 1]

<p><span lang="EN-IN"> &lt;!-- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1820669517"> </o:OLEObject>&lt;/xml&gt;&lt;![endif]--&gt;</span></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>,</mo> </mtd> <mtd columnalign="center"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>,</mo> </mtd> <mtd columnalign="center"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mrow> </math> </span></span></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mi>x</mi> <mo>,</mo> </mtd> <mtd columnalign="center"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mo>−</mo> <mi>x</mi> <mo>,</mo> </mtd> <mtd columnalign="center"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </mrow> </math> </span>&nbsp; -&gt;</span><span lang="EN-IN">g(x) = |x|, x </span><span lang="EN-IN">Î</span><span lang="EN-IN"> (–3, 1)</span></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>,</mo> </mtd> <mtd columnalign="center"> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>⇒</mo> <mi>x</mi> <mo>∈</mo> <mi>?</mi> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>,</mo> </mtd> <mtd columnalign="center"> <mrow> <mo>|</mo> <mrow> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⇒</mo> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&lt;!-- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1820669519"> </o:OLEObject>&lt;/xml&gt;&lt;![endif]--&gt;</span></p><p><span lang="EN-IN"> &lt;!-- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1820669527"> </o:OLEObject>&lt;/xml&gt;&lt;![endif]--&gt;</span></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>,</mo> </mtd> <mtd columnalign="center"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>,</mo> </mtd> <mtd columnalign="center"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </mrow> </math> </span></span></p><div><div><picture><source srcset="" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1759144079phpDRabfs.jpeg" alt="" width="265" height="129" loading="lazy"></picture></div></div><p><span lang="EN-IN">Range of fog(x) is [0, 1]</span></p><p><span lang="EN-IN">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</span></p><p><span lang="EN-IN">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Range of fog(x) is [0, 1]</span></p>

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