Graph the function. [tex]f(x)=\sqrt{x+1}[/tex]
Answer (2)
Determine the domain of the function f ( x ) = x + 1 by solving x + 1 ≥ 0 , which gives x ≥ − 1 .
Find the x-intercept by setting f ( x ) = 0 , resulting in x = − 1 .
Calculate additional points on the graph, such as ( 0 , 1 ) , ( 3 , 2 ) , and ( 8 , 3 ) .
Sketch the graph starting from ( − 1 , 0 ) , showing an increasing curve with a decreasing rate.
The graph of the function f ( x ) = x + 1 is a square root function shifted one unit to the left, starting at ( − 1 , 0 ) .
Explanation
Understanding the Function We are asked to graph the function f ( x ) = x + 1 . This is a square root function, which means the expression inside the square root must be non-negative.
Determining the Domain To find the domain, we solve the inequality x + 1 ≥ 0 , which gives us x ≥ − 1 . Thus, the domain of the function is [ − 1 , ∞ ) .
Finding the X-Intercept To find the x-intercept, we set f ( x ) = 0 and solve for x : x + 1 = 0 Squaring both sides, we get x + 1 = 0 , so x = − 1 . The x-intercept is ( − 1 , 0 ) .
Finding Additional Points Let's find a few additional points on the graph to help us sketch it. We'll choose x values greater than or equal to -1.
If x = 0 , then f ( 0 ) = 0 + 1 = 1 = 1 . So, the point ( 0 , 1 ) is on the graph. If x = 3 , then f ( 3 ) = 3 + 1 = 4 = 2 . So, the point ( 3 , 2 ) is on the graph. If x = 8 , then f ( 8 ) = 8 + 1 = 9 = 3 . So, the point ( 8 , 3 ) is on the graph.
Sketching the Graph Now we plot the points ( − 1 , 0 ) , ( 0 , 1 ) , ( 3 , 2 ) , and ( 8 , 3 ) on a coordinate plane. Starting at the point ( − 1 , 0 ) , we sketch the graph of the function, which increases as x increases, but at a decreasing rate. The graph looks like a square root function shifted one unit to the left.
Final Answer The graph of f ( x ) = x + 1 starts at ( − 1 , 0 ) and increases gradually as x increases.
Examples
Square root functions are used in various real-world applications, such as calculating the speed of a wave, determining the period of a pendulum, or modeling growth rates. For example, if you're designing a suspension bridge, you might use a square root function to calculate the cable sag based on the length of the span and the tension in the cables. Understanding the behavior of square root functions helps engineers ensure the structural integrity and safety of the bridge. The formula v = g r relates the maximum velocity v with which a car can safely turn on a curved road to the radius r of the curve and the gravitational acceleration g .
The graph of the function f ( x ) = x + 1 starts at the point ( − 1 , 0 ) and increases for all x ≥ − 1 . It is defined for the domain [ − 1 , ∞ ) and consists of a square root function that rises gradually. Key points on the graph include ( 0 , 1 ) , ( 3 , 2 ) , and ( 8 , 3 ) .
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