Find the range of the function f ( x ) = x 2 - 2 for x ∈ ℝ

Given f ( x ) = x 2 - 2 for x ∈ ℝ The part of the expression is a square so it will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 0 . Therefore, when x = 0 then f x = - 2 When x = 2 then f x = 2 2 - 2 = 2 The minimum value of the function is - 2 The set of output values for a function is called the range of the function. If f x ≥ a then a is called the minimum value of f x and f x belongs to set of all real numbers which are greater than or equal to a . Hence, the range is f x ≥ - 2

Find the range of the function f ( x ) = x 2 - 2 for x ∈ ℝ

Given f ( x ) = x 2 - 2 for x ∈ ℝ The part of the expression is a square so it will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 0 . Therefore, when x = 0 then f x = - 2 When x = 2 then f x = 2 2 - 2 = 2 The minimum value of the function is - 2 The set of output values for a function is called the range of the function. If f x ≥ a then a is called the minimum value of f x and f x belongs to set of all real numbers which are greater than or equal to a . Hence, the range is f x ≥ - 2

Find the range of the function f x = 3 - 2 x 2 for x ≤ 2

Given f x = 3 - 2 x 2 for x ≤ 2 The part of the expression is a square so it will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 0 . The maximum value of f x occurs when x is minimum Therefore, when x = 0 then f x = 3 is maximum When x = 2 then f x = 3 - 2 ( 2 ) 2 = - 5 When x = - 1 then f x = 3 - 2 - 1 2 = 1 When x = - 2 then f x = 3 - 2 - 2 2 = - 5 When x = - 5 then f x = 3 - 2 - 5 2 = - 47 If f x ≤ b then b is called the maximum value of f x . and f x belongs to set of all real numbers which are less than or equal to b . Hence, the range is f x ≤ 3

Find the range of the function f ( x ) = ( x - 2 ) 2 + 5 for x ≥ 2

Given f ( x ) = ( x - 2 ) 2 + 5 for x ≥ 2 The part of the expression is a square so it will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 2 . The minimum value of f x occurs when x = 2 Therefore, when x = 2 then f x = 5 is minimum When x = 4 then f x = 4 - 2 2 + 5 = 9 From the above values we can observe f x increases when x increases with x ≥ 2 If f x ≥ a then a is called the minimum value of f x and f x is the set of all real numbers. Hence, the range is f x ≥ 5

Find the range of the function f ( x ) = ( 2 x - 1 ) 2 - 7 for x ≥ 1 2

Given f ( x ) = ( 2 x - 1 ) 2 - 7 for x ≥ 1 2 The part of the expression is a square so it will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 1 2 . The minimum value of f x occurs when x = 1 2 Therefore, when x = 1 2 then f x = 2 × 1 2 - 1 - 7 = - 7 is minimum When x = 4 then f x = 8 - 1 2 - 7 = 42 From the above values we can observe f x increases when x increases with x ≥ 1 2 If f x ≥ a then a is called the minimum value of f x and f x is the set of all real numbers. Hence, the range is f x ≥ - 7

Find the range of the function f x = 8 - ( x - 5 ) 2 for 4 ≤ x ≤ 10

Given f x = 8 - ( x - 5 ) 2 for 4 ≤ x ≤ 10 The part of the expression is a square so it will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 5 . The maximum value of f x occurs when x = 5 Therefore, when x = 5 then f x = 8 - 5 - 5 2 = 8 is maximum When x = 4 then f x = 8 - 4 - 5 2 = 7 When x = 7 then f x = 8 - 7 - 5 2 = 4 When x = 10 then f x = 8 - 10 - 5 2 = - 17 From the above values we can observe f x decreases when x increases with 4 ≤ x ≤ 10 If a ≤ f x ≤ b then a is called the minimum value of f x and b is called the maximum value of f x , f x is the set of all real numbers lies between a and b Hence, the range is - 17 ≤ f x ≤ 8

Find the range of the function f ( x ) = 1 + x - 4 for x ≥ 4 .

Given f ( x ) = 1 + x - 4 for x ≥ 4 The expression under square root will always be ≥ 0 . The smallest value it can be is 0 . This occurs when x = 4 . The minimum value of f x occurs when x = 4 Therefore, when x = 4 then f x = 1 + 4 - 4 = 1 is minimum When x = 5 then f x = 1 + 5 - 4 = 2 From the above values we can observe f x increases when x increases with x ≥ 4 If f x ≥ a then a is called the minimum value of f x , f x is the set of all real numbers. Hence, the range is f x ≥ 1 .