For the function [tex]f(x)=\sqrt{x+1}[/tex],
a) State the domain of the function, using interval notation.
Domain of [tex]f(x)[/tex] : [ ]
b) Graph the function
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c) Use the graph to determine the range of the function. Use interval notation.
Answer (2)
Find the domain by solving the inequality x + 1 ≥ 0 , which gives x ≥ − 1 .
Express the domain in interval notation as [ − 1 , ∞ ) .
Determine the range by considering the possible values of x + 1 , which are all non-negative numbers.
Express the range in interval notation as [ 0 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = x + 1 . We need to find the domain and range of this function and also graph it.
Finding the Domain To find the domain, we need to determine the set of all possible x values for which the function is defined. Since we have a square root, the expression inside the square root must be non-negative. Therefore, we need to solve the inequality x + 1 ≥ 0 .
Expressing the Domain Solving the inequality x + 1 ≥ 0 , we subtract 1 from both sides to get x ≥ − 1 . In interval notation, this is [ − 1 , ∞ ) .
Graphing the Function The graph of the function f ( x ) = x + 1 starts at the point ( − 1 , 0 ) and increases as x increases. It is a transformation of the basic square root function y = x , shifted one unit to the left.
Determining the Range To find the range, we need to determine the set of all possible y values that the function can take. Since the square root function always returns non-negative values, and the smallest value of x + 1 is 0 (when x = − 1 ), the smallest value of f ( x ) is 0 = 0 . As x increases, f ( x ) also increases without bound. Therefore, the range is [ 0 , ∞ ) .
Final Answer The domain of the function f ( x ) = x + 1 is [ − 1 , ∞ ) , and the range is [ 0 , ∞ ) .
Examples
Consider a scenario where you are tracking the distance a car can travel after filling its gas tank, where the distance depends on the amount of fuel in the tank. If the distance d ( f ) is given by d ( f ) = f + 1 , where f is the amount of fuel (in gallons) above a reserve level, then understanding the domain and range helps determine the possible fuel levels and corresponding distances. For instance, the car can only travel a real distance if f ≥ − 1 , meaning the fuel level must be at least at the reserve. The range tells us the possible distances the car can travel, starting from 0 when f = − 1 .
The domain of the function f ( x ) = x + 1 is [ − 1 , ∞ ) and the range is [ 0 , ∞ ) .
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