What is the range of the function $y=\sqrt{x+5}$?
A. $y \geq-5$
B. $y \geq 0$
C. $y \geq \sqrt{5}$
D. $y \geq 5
Answer (2)
Determine the domain of the function by ensuring the expression inside the square root is non-negative: x + 5 ≥ 0 , which gives x ≥ − 5 .
Recognize that the square root function always returns non-negative values: y = x + 5 ≥ 0 .
Find the minimum value of y by using the smallest value in the domain, x = − 5 , which gives y = − 5 + 5 = 0 .
Conclude that the range of the function is all non-negative values: y ≥ 0 .
Explanation
Understanding the problem We are asked to find the range of the function y = x + 5 . The range of a function is the set of all possible output values (y-values) that the function can produce.
Domain of the function The square root function, denoted by , is only defined for non-negative values. This means that the expression inside the square root must be greater than or equal to zero. In our case, we must have x + 5 ≥ 0
Finding the domain Solving the inequality x + 5 ≥ 0 for x , we subtract 5 from both sides: x ≥ − 5 This tells us that the domain of the function is all x values greater than or equal to -5.
Finding the range Now, let's consider the output of the square root function. The square root of a non-negative number is always non-negative. Therefore, y = x + 5 will always be greater than or equal to zero.
To find the minimum value of y , we can use the smallest value in the domain, which is x = − 5 . When x = − 5 , we have y = − 5 + 5 = 0 = 0 Since x can take any value greater than or equal to -5, and the square root function is monotonically increasing (i.e., as x increases, y also increases), y can take any non-negative value.
Final Answer Therefore, the range of the function y = x + 5 is y ≥ 0 .
Examples
Imagine you are designing a garden, and you want to determine the possible lengths of a side of a square garden bed. If the area of the garden bed is represented by the function A = s 2 , where s is the side length, then the side length can be found by s = A . Since the area must be non-negative, the side length will also be non-negative. Understanding the range of the square root function helps you determine the possible side lengths for different areas.
The range of the function y = x + 5 is determined by the square root's properties, leading us to conclude that all output values must be non-negative. Therefore, the range is y ≥ 0 . The answer is B. y ≥ 0 .
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