Graph the function $f(x)=\sqrt{x-1}$. Choose the correct graph below.
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Answer (2)
Determine the domain of the function f ( x ) = x − 1 : x ≥ 1 .
Find the x-intercept by setting f ( x ) = 0 : x = 1 .
Check for a y-intercept by setting x = 0 : No y-intercept exists.
Conclude that the graph starts at ( 1 , 0 ) and increases to the right, which corresponds to the correct graph.
Explanation
Understanding the Function The function we need to graph is f ( x ) = s q r t x − 1 . We need to identify the correct graph from the options provided.
Finding the Domain First, let's determine the domain of the function. Since we have a square root, the expression inside the square root must be non-negative: x − 1 g e q 0 x g e q 1 So, the domain of the function is x g e q 1 . This means the graph will start at x = 1 and extend to the right.
Finding the X-Intercept Next, let's find the x-intercept. To find the x-intercept, we set f ( x ) = 0 :
s q r t x − 1 = 0 x − 1 = 0 x = 1 So, the x-intercept is at x = 1 .
Finding the Y-Intercept Now, let's find the y-intercept. To find the y-intercept, we set x = 0 :
f ( 0 ) = s q r t 0 − 1 = s q r t − 1 Since we cannot take the square root of a negative number in the real number system, there is no y-intercept.
Determining the Shape of the Graph The general shape of the graph of f ( x ) = s q r t x is a curve that starts at the origin and increases as x increases. The graph of f ( x ) = s q r t x − 1 is a horizontal shift of the graph of f ( x ) = s q r t x one unit to the right.
Identifying the Correct Graph Based on our analysis:
The domain is x g e q 1 .
The x-intercept is at x = 1 .
There is no y-intercept.
The graph starts at x = 1 and increases to the right.
Therefore, the correct graph is the one that starts at ( 1 , 0 ) and increases to the right.
Examples
Square root functions are used in various real-world applications, such as calculating the distance between two points, determining the speed of an object in free fall, or modeling the growth of certain populations. For example, if you want to determine the length of the diagonal of a square with area A, you would use the square root function: diagonal = 2 A . Understanding the properties and graphs of square root functions helps in solving these practical problems.
The graph of the function f ( x ) = x − 1 starts at the point (1, 0) and increases to the right, with a domain of x ≥ 1 and no y-intercept. The correct graph option is the one that represents this behavior. Therefore, the answer is based on visual confirmation of the graph that follows these characteristics.
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