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Range of a function Quiz 2

mathematics

 If $f\left( x \right)=a{{x}^{2}}+bx+c$ and $g\left( x \right)=3$, then range of $g\circ f(x)$ is:

 Open the composition by substituting their respective values.

  Since, $g\circ f(x)=g\left( a{{x}^{2}}+bx+c \right)$ and $g\left( x \right)$ is a constant function so, the value of $g\left( x \right)$ will always be 3, irrespective of $x$. So, $g\circ f(x)=3$ gives the range of $g\circ f(x)$ is 3.

 A small boy of 5years old was very fond of rides. Every year in his village fair, he used to ride a lot of rides. He got a pamphlet of a park which was going to open after a few months. He saw a ride whose schema is as per below curve: There was a precaution that for less than 8 yr old child, the height he can go to = $\dfrac{Maxm.\text{ Height}}{2}$. Find the range of height,  a 7 yr old child will travel on the ride?  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/26f4f8ab-cba7-40da-8eda-5c4ef97f613e1918430725064415508.png"/>

 Closed bracket means the boundary value is also attainable by the function.   

  $\text{From the graph, we can see that the:} \\ \text{Maximum Height= 100} \\ \text{As the boy is less than 8yr old i.e, 7yr old, } \\ \text{So, as per question, he can start from a maxm. height of } \\ =\dfrac{Maxm.\text{ Height}}{2}=\dfrac{100}{2}=50 \\ \text{Lowest point he will travel = -50, derived from the graph} \\ \text{Thus, the range of height a 7yr old child will travel = [50,-50].} $ Reason of closed bracket is that he will surely go to that specific height thus the closed bracket needs to be used.

 Find the range of the function given below: $y=\left| \left| \left| \sin \sqrt{2{{x}^{6}}+4{{x}^{8}}+5} \right| \right|.....\infty times \right|-1$

 Mod converts negative value to positive value  

  $\text{Assume }\sqrt{2{{x}^{6}}+4{{x}^{8}}+5}=t \\ \text{Now, (sin t) have range from -1 to 1 while }\left| \sin \text{ t} \right|\text{ have range 0 to 1.} \\ \text{As }\left| \sin \text{ t} \right|\text{ is positive throughout the graph} \\  \text{Thus the next }\infty \text{ mod will have no effect on it.} \\ \text{Thus range of } \\ \left| \left| \left| \sin \sqrt{2{{x}^{6}}+4{{x}^{8}}+5} \right| \right|.....\infty\text{ times }\right|\text{ is 0 to 1 only. }\\ \text{Range of y =} (0-1)\text{ to }(1-1)= -1\text{ to }0 = [-1,0] $

 A lost man is standing between two cliffs. The angle of elevation of one cliff to his right.  from the point where he stands is $30{}^\circ $ and the angle of depression from the top of the other cliff  which is to his left, to the point where man is standing is ${{45}^{\circ }}$. He wants to shoot  the SOS gun so that someone can help him. What is the available range of angle for shooting, when measured from the right hand side?

 Find angle between base of the two cliffs and solve accordingly to get the range. 

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/9049550b-debb-44da-b741-39ec88ccbccf7664713728161767602.png"/>$\\ \text{Angle left = }{{180}^{\circ }}-{{45}^{\circ }}-{{30}^{\circ }}={{105}^{\circ }} \\ \text{Starting Angle = }\text{3}{{\text{0}}^{\text{o}}}\\ \text{Ending Angle = }{{30}^{\circ }}+{{105}^{\circ }}={{135}^{\circ }} \\ \text{Therefore, available angle = }({{30}^{\circ }},{{135}^{\circ }})$ 

 Find the value of x: $\left| x+1 \right|\le 4;\left| x+1 \right|\ge 4$

 Assume $\left| x+1 \right|$ as a number as P.

  $\text{Let’s assume, }\left| x+1 \right|\text{ as a number as P}. \\ \text{As per given eqn.} \\ P\le 4\text{ , P}\ge \text{4} \\ \text{Therefore to satisfy both condition, }P=4 \\ \left| x+1 \right|=4 \\ x+1=\pm 4 \\ \text{Case 1:} \\ x+1=4 \\ x=3 \\ \\ \text{Case 2:} \\ x+1=-4 \\ x=-5 \\ \text{Therefore, }x= (-5), 3$