For the function [tex]f(x)=\sqrt{x-5}[/tex],
a) State the domain of the function, using interval notation.
Domain of [tex]f(x)[/tex] : [$\square$]
b) Graph the function
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c) Use the graph to determine the range of the function. Use interval notation.
Range of [tex]f(x)[/tex] : [$\square$]
Answer (1)
Determine the domain by solving the inequality x − 5 ≥ 0 , which gives x ≥ 5 .
Express the domain in interval notation as [ 5 , ∞ ) .
Determine the range by recognizing that the square root function returns non-negative values, so f ( x ) ≥ 0 .
Express the range in interval notation as [ 0 , ∞ ) .
The domain of f ( x ) is [ 5 , ∞ ) . The range of f ( x ) is [ 0 , ∞ ) .
Explanation
Understanding the problem We are given the function f ( x ) = s q r t x − 5 and we need to find its domain and range, and also graph it.
Finding the domain To find the domain of the function, we need to determine the values of x for which the function is defined. Since we have a square root, the expression inside the square root must be greater than or equal to 0. Therefore, we need to solve the inequality: x − 5 ≥ 0
Expressing the domain in interval notation Adding 5 to both sides of the inequality, we get: x ≥ 5
In interval notation, the domain is [ 5 , ∞ ) .
Finding the range To find the range, we need to determine the possible values of f ( x ) . Since the square root function always returns non-negative values, f ( x ) ≥ 0 . As x varies within the domain [ 5 , ∞ ) , f ( x ) will take on all non-negative values. Therefore, the range is [ 0 , ∞ ) .
Graphing the function The graph of the function f ( x ) = x − 5 is a square root function that starts at the point ( 5 , 0 ) and increases as x increases.
Final Answer In summary:
The domain of f ( x ) is [ 5 , ∞ ) .
The range of f ( x ) is [ 0 , ∞ ) .
Examples
Understanding the domain and range of functions is crucial in many real-world applications. For example, if we are modeling the distance a car can travel based on the amount of fuel it has, we need to ensure that the amount of fuel is non-negative and within the car's tank capacity. Similarly, when calculating the height of a projectile, we need to consider that height cannot be negative. These constraints define the domain and range of the functions used in these models, ensuring that our calculations are meaningful and accurate.
