Select the correct answer.
What is the range of function [tex]$m$[/tex]?
[tex]$m(x)=\sqrt{x-3}+1$[/tex]
A. [tex]$(-\infty, 3]$[/tex]
B. [tex]$(-\infty, 1]$[/tex]
C. [tex]$[3, \infty)$[/tex]
D. [tex]$[1, \infty)$[/tex]
Answer (2)
The range of the function m ( x ) = x − 3 + 1 is determined to be [ 1 , ∞ ) since the minimum value occurs at x = 3 and is equal to 1. As x increases, the function continues to grow without an upper limit. Therefore, the correct answer is D: [ 1 , ∞ ) .
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Determine the domain of the function m ( x ) = x − 3 + 1 , which is x ≥ 3 .
Find the minimum value of the function by evaluating m ( 3 ) = 1 .
Observe that as x increases, m ( x ) also increases without bound.
Conclude that the range of the function is [ 1 , ∞ ) .
[ 1 , ∞ )
Explanation
Understanding the Function Let's analyze the function m ( x ) = x − 3 + 1 to determine its range. The range of a function is the set of all possible output values (y-values) that the function can produce.
Determining the Domain First, we need to consider the domain of the function. Since we have a square root, the expression inside the square root must be non-negative:
x − 3 ≥ 0
Solving for x , we get:
x ≥ 3
So, the domain of the function is [ 3 , ∞ ) .
Finding the Minimum Value Now, let's find the minimum value of the function. The smallest value of x in the domain is x = 3 . When x = 3 , we have:
m ( 3 ) = 3 − 3 + 1 = 0 + 1 = 0 + 1 = 1
Since the square root function always returns a non-negative value, the minimum value of x − 3 is 0. Therefore, the minimum value of m ( x ) is 1.
Determining the Upper Bound Next, we need to determine if there is an upper bound for the function. As x increases, the value of x − 3 also increases. Consequently, x − 3 increases as well. Since there is no upper limit to how large x can be (as x approaches infinity), the value of x − 3 can also become infinitely large. Therefore, m ( x ) = x − 3 + 1 can also become infinitely large.
Concluding the Range Since the minimum value of m ( x ) is 1 and it can increase without bound, the range of the function is all real numbers greater than or equal to 1. In interval notation, this is [ 1 , ∞ ) . Therefore, the correct answer is D.
Examples
Imagine you are tracking the height of a plant over time. The function m ( x ) = x − 3 + 1 could represent the plant's height (in inches) after x days, starting from day 3 (since the plant needs a few days to sprout). The range of this function tells you the possible heights the plant can reach. For example, the plant will always be at least 1 inch tall, and it can grow indefinitely, which is why the range is [ 1 , ∞ ) . Understanding the range helps you set expectations for the plant's growth and plan accordingly.
