Which points lie on the graph of [tex]f(x)=-\sqrt{-x}[/tex]?
| x | y |
|---|---|
| 3 | -2 |
| | |
| | |
[tex](-9,3)[/tex]
[tex](-4,-2)[/tex]
[tex](-1,1)[/tex]
[tex](0,0)[/tex]
[tex](1,1)[/tex]
[tex](2,-4)[/tex]
[tex](9,3)[/tex]
Answer (1)
Check if each point ( x , y ) satisfies the equation y = − − x .
For ( − 9 , 3 ) : f ( − 9 ) = − 3 = 3 .
For ( − 4 , − 2 ) : f ( − 4 ) = − 2 , so ( − 4 , − 2 ) lies on the graph.
For ( − 1 , 1 ) : f ( − 1 ) = − 1 = 1 .
For ( 0 , 0 ) : f ( 0 ) = 0 , so ( 0 , 0 ) lies on the graph.
Points ( 1 , 1 ) , ( 2 , − 4 ) , and ( 9 , 3 ) are not in the domain of the function.
The points on the graph are: ( − 4 , − 2 ) , ( 0 , 0 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = − − x and a set of points. Our goal is to determine which of these points lie on the graph of the function. A point ( x , y ) lies on the graph if and only if y = f ( x ) . Also, we need to consider the domain of the function, which is x ≤ 0 .
Checking Each Point Let's check each point:
Point (-9, 3): We need to check if 3 = f ( − 9 ) .
f ( − 9 ) = − − ( − 9 ) = − 9 = − 3 Since 3 = − 3 , the point ( − 9 , 3 ) does not lie on the graph.
Point (-4, -2): We need to check if − 2 = f ( − 4 ) .
f ( − 4 ) = − − ( − 4 ) = − 4 = − 2 Since − 2 = − 2 , the point ( − 4 , − 2 ) lies on the graph.
Point (-1, 1): We need to check if 1 = f ( − 1 ) .
f ( − 1 ) = − − ( − 1 ) = − 1 = − 1 Since 1 = − 1 , the point ( − 1 , 1 ) does not lie on the graph.
Point (0, 0): We need to check if 0 = f ( 0 ) .
f ( 0 ) = − − 0 = − 0 = 0 Since 0 = 0 , the point ( 0 , 0 ) lies on the graph.
Point (1, 1): Since 0"> 1 > 0 , the function f ( x ) = − − x is not defined for x = 1 . Therefore, the point ( 1 , 1 ) does not lie on the graph.
Point (2, -4): Since 0"> 2 > 0 , the function f ( x ) = − − x is not defined for x = 2 . Therefore, the point ( 2 , − 4 ) does not lie on the graph.
Point (9, 3): Since 0"> 9 > 0 , the function f ( x ) = − − x is not defined for x = 9 . Therefore, the point ( 9 , 3 ) does not lie on the graph.
Final Answer The points that lie on the graph of f ( x ) = − − x are ( − 4 , − 2 ) and ( 0 , 0 ) .
Examples
Understanding which points lie on a graph is crucial in many real-world applications. For instance, in physics, you might have a function that describes the trajectory of a projectile. Knowing which points lie on this trajectory helps you predict where the projectile will be at a given time. Similarly, in economics, a demand curve shows the relationship between the price of a product and the quantity demanded. Identifying points on this curve allows businesses to make informed decisions about pricing and production levels. This skill is fundamental in data analysis, where you often need to verify if data points fit a particular model or function.
