What is the range of function [tex]g[/tex]?
[tex]g(x)=\sqrt{x-1}+2[/tex]
A. [tex]y \geq 2[/tex]
B. [tex]y \leq 2[/tex]
C. [tex]y \geq 1[/tex]
D. [tex]y \leq 1[/tex]
Answer (1)
Determine the domain of the function: x ≥ 1 .
Find the minimum value of the function: g ( 1 ) = 2 .
Analyze the behavior of the function: as x increases, g ( x ) increases.
Conclude the range of the function: y ≥ 2 .
Explanation
Understanding the Problem We are given the function g ( x ) = s q r t x − 1 + 2 and asked to find its range. The range of a function is the set of all possible output values (y-values).
Finding the Domain First, we need to determine the domain of the function. Since we have a square root, the expression inside the square root must be non-negative: x − 1 g e q 0 Solving for x , we get: x g e q 1 So, the domain of the function is x g e q 1 .
Determining the Minimum Value Now, let's analyze the behavior of the function. The square root function x − 1 always returns a non-negative value. The smallest value of x − 1 is 0, which occurs when x = 1 . Therefore, the smallest value of g ( x ) is: g ( 1 ) = \[ s q r t 1 − 1 + 2 = 0 + 2 = 2
Finding the Range As x increases, x − 1 also increases. Therefore, g ( x ) increases as well. Since the minimum value of g ( x ) is 2 and g ( x ) can take any value greater than or equal to 2, the range of g ( x ) is y g e q 2 .
Final Answer Therefore, the range of the function g ( x ) = \[ s q r t x − 1 + 2 is y g e q 2 . The correct answer is A.
Examples
Understanding the range of a function is crucial in many real-world applications. For example, if g ( x ) represents the daily production cost of a factory where x is the number of items produced, knowing the range tells us the minimum cost the factory will incur each day. If the function is g ( x ) = \[ s q r t x − 1 + 2 , it means the factory will always have a minimum cost of 2 ( y g e q 2 ) , regardless of how many items are produced. This kind of analysis helps in budgeting and planning.
