Let X = { 3 , 4 , 6 , 8 } . Determine whether the relation R = ( x , f ( x ) ) ∣ x ∈ X , f ( x ) = x 2 + 1 is a function from X to N ?

Given, x = { 3 , 4 , 6 , 8 } and relation R = ( x , f ( x ) ) ∣ x ∈ X , f ( x ) = X 2 + 1 f ( x ) = x 2 + 1 f ( 3 ) = 3 2 + 1 = 10 f ( 4 ) = 4 2 + 1 = 17 f ( 6 ) = 6 2 + 1 = 37 f ( 8 ) = 8 2 + 1 = 65 A function is a special type of relation in which each element of the domain is paired with exactly one element in the range. A mapping shows how the elements are paired. Its like a flow chart for a function, showing the input and output values. The pictorial representation of the above relation is R = { ( 3 , 10 ) , ( 4 , 17 ) , ( 6 , 37 ) , ( 8 , 65 ) } Here, for x only one value exists of f x in all natural number N Hence, yes R is a function from X to N .

Given the function f : x → x 2 - 5 x + 6 , evaluate f ( - 1 )

Given, function is f ( x ) = x 2 - 5 x + 6 A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. To evaluate put x = - 1 in function So, f ( - 1 ) = ( - 1 ) 2 - 5 ( - 1 ) + 6 = 1 + 5 + 6 = 12 Thus, the required value of f - 1 is 12 .

Given the function f : x → x 2 - 5 x + 6 , evaluate f ( 2 a )

Given, function is f ( x ) = x 2 - 5 x + 6 A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. To evaluate put x = 2 a in function So, f ( 2 a ) = ( 2 a ) 2 - 5 ( 2 a ) + 6 = 4 a 2 - 10 a + 6 Thus, the required value of f 2 a is 4 a 2 - 10 a + 6 .

Given the function f : x → x 2 - 5 x + 6 , evaluate f ( 2 )

Given, function is f ( x ) = x 2 - 5 x + 6 A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. To evaluate put x = 2 in the function So, f ( 2 ) = 2 2 - 5 ( 2 ) + 6 = 4 - 10 + 6 = 0 Thus, the required value of f 2 is 0 .

Given the function f : x → x 2 - 5 x + 6 , evaluate f ( x - 1 ) .

Given, function is f ( x ) = x 2 - 5 x + 6 . A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. To evaluate f x - 1 , put x = x - 1 in the function So, f ( x - 1 ) = ( x - 1 ) 2 - 5 ( x - 1 ) + 6 = x 2 - 2 x + 1 - 5 x + 5 + 6 = x 2 - 7 x + 12 Thus, the required value of f x - 1 is x 2 - 7 x + 12 .

A graph representing the function f ( x ) is given in Figure, it is clear that f ( 9 ) = 2 . Find the following value of the function f ( 0 )

A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. The given graph is For f 0 , check the value corresponding to x = 0 So, f ( 0 ) = 9 Thus, the required value of f 0 is 9 .