Sketch the graph of the curve y = sin x + π 2 .

We have y = sin x + π 2 . We know the graph of y = sin x + π 2 is obtained by shifting the graph of y = sin x by π 2 units towards left along the x -axis. Hence, the graph of y = sin x + π 2 is given below

Sketch the graph of the curve y = - x , ( where · represents the fractional part function )

We have y = - x , ( where · represents the fractional part function ) . We know that, the value of x lies between [ 0 , 1 ) . Hence, the value of - x lies between ( - 1 , 0 ] . Thus, the graph of y = - x is given below .

Sketch the graph of the curve y = x + x - x , ( where · denotes the greatest integer function ) .

Given y = x + x - x . We know that x - x = x , where · denotes the greatest integer function and · denotes the fractional part function. Thus, the function become y = x + x . We know that, the range of the fractional part function i.e. x is [ 0 , 1 ) , hence, the range of x is also [ 0 , 1 ) . And, the range of the greatest integer function i.e. x is all integers. Hence, x = ⋮ ⋮ - 1 , - 1 ≤ x < 0 0 , 0 ≤ x < 1 1 , 1 ≤ x < 2 2 , 2 ≤ x < 3 ⋮ ⋮ . Thus, the value of the function y = x + x is y = ⋮ ⋮ - 1 + x , - 1 ≤ x < 0 0 + x , 0 ≤ x < 1 1 + x , 1 ≤ x < 2 2 + x , 2 ≤ x < 3 ⋮ ⋮ . Hence, the graph of the function y = x + x is given below,

Sketch the graph of the curve y = 2 - x 2 .

We know that the square root of a positive number is always positive and the function under square root is defined for non-negative values. Hence, the value y = 2 - x 2 is always positive and the function is defined for 2 - x 2 ≥ 0 , ⇒ x 2 ≤ 2 ⇒ - 2 ≤ x ≤ 2 . Also, we know that the graph of - x 2 is a parabola open downward and vertex at origin. The graph of y = f x + k , k > 0 is obtained by shifting the graph of y = f x , k units upwards along the y -axis. Hence, the graph of 2 - x 2 is also a parabola open downward and vertex at 0 , 2 . Thus, the graph of y = 2 - x 2 is given below

E M B I B E

Gamma Question Bank for Engineering Mathematics>Class XI>Chapter 2 - Relations and Functions>EXERCISE 2.5

Embibe Experts Solutions for Exercise 5: EXERCISE 2.5

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Embibe Experts Mathematics Solutions for Exercise - Embibe Experts Solutions for Exercise 5: EXERCISE 2.5

Attempt the free practice questions from Exercise 5: EXERCISE 2.5 with hints and solutions to strengthen your understanding. Gamma Question Bank for Engineering Mathematics solutions are prepared by Experienced Embibe Experts.