At which root does the graph of [tex]f(x)=(x+4)^0(x+7)^0[/tex] cross the [tex]x[/tex]-axis?
A. -7
B. -4
C. 4
D. 7
Answer (2)
The function f ( x ) = ( x + 4 ) 0 ( x + 7 ) 0 simplifies to 1 for all x except -4 and -7, meaning it does not cross the x-axis. Therefore, the graph does not have any roots. The question is based on a flawed premise since the graph does not intersect the x-axis at any point.
;
Simplify the function: f ( x ) = ( x + 4 ) 0 ( x + 7 ) 0 = 1 for x = − 4 , − 7 .
Analyze the graph: The graph is a horizontal line y = 1 with holes at x = − 4 and x = − 7 .
Determine the roots: The graph does not cross the x-axis, so there are no roots.
Conclude: The question is ill-posed, but if it refers to ( x + 4 ) ( x + 7 ) , the roots are -4 and -7. Since the function as defined does not cross the x-axis, the question is flawed. There is no root where the graph crosses the x-axis. However, if we were to consider a modified function like g ( x ) = ( x + 4 ) ( x + 7 ) , the roots would be -4 and -7. Since the given options are -7, -4, 4, and 7, and the function as defined does not cross the x-axis, the question is ill-posed. However, if we assume the question meant to ask about the roots of ( x + 4 ) ( x + 7 ) , we can choose either -4 or -7. Since the function is not defined at x = − 4 and x = − 7 , and the graph does not cross the x-axis at any point, there is technically no answer to the question. However, if we are forced to choose from the given options, and assuming the question meant to ask about the roots of ( x + 4 ) ( x + 7 ) , we can choose either -4 or -7. − 7
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 4 ) 0 ( x + 7 ) 0 and asked to find where its graph crosses the x -axis. This means we are looking for the roots of the function, i.e., the values of x for which f ( x ) = 0 .
Simplifying the Function First, let's simplify the function. Recall that any non-zero number raised to the power of 0 is 1. Therefore, ( x + 4 ) 0 = 1 when x e q − 4 and ( x + 7 ) 0 = 1 when x e q − 7 . Thus, for x e q − 4 and x e q − 7 , we have f ( x ) = ( 1 ) ( 1 ) = 1 .
Analyzing the Function The function f ( x ) is equal to 1 everywhere except at x = − 4 and x = − 7 . At these points, the function is undefined because 0 0 is undefined. However, for the sake of this problem, we can consider 0 0 = 1 . So, the function is f ( x ) = 1 for all x .
Finding the Roots Since f ( x ) = 1 for all x , the graph of f ( x ) is a horizontal line at y = 1 . This line never intersects the x -axis, which is the line y = 0 . Therefore, the graph of f ( x ) does not cross the x -axis at any point.
Conclusion Since the graph of f ( x ) never crosses the x -axis, there are no roots. Therefore, the question is based on a flawed premise. However, if the question intended to ask about the roots of g ( x ) = ( x + 4 ) ( x + 7 ) , then the roots would be x = − 4 and x = − 7 . But that is not the function we were given.
Final Answer Given the function f ( x ) = ( x + 4 ) 0 ( x + 7 ) 0 , the graph does not cross the x-axis. Therefore, there is no root where the graph crosses the x-axis. However, if we were to consider a modified function like g ( x ) = ( x + 4 ) ( x + 7 ) , the roots would be -4 and -7. Since the given options are -7, -4, 4, and 7, and the function as defined does not cross the x-axis, the question is ill-posed. However, if we assume the question meant to ask about the roots of ( x + 4 ) ( x + 7 ) , then the answer would be either -4 or -7. Since the function is not defined at x = − 4 and x = − 7 , and the graph does not cross the x-axis at any point, there is technically no answer to the question. However, if we are forced to choose from the given options, and assuming the question meant to ask about the roots of ( x + 4 ) ( x + 7 ) , we can choose either -4 or -7.
Examples
Understanding roots of functions is crucial in many real-world applications. For instance, in physics, the roots of a function describing the trajectory of a projectile can tell us when the projectile hits the ground. In engineering, roots can represent equilibrium points in a system. In economics, roots can represent break-even points where costs equal revenue. The function f ( x ) = ( x + a ) ( x + b ) has roots at x = − a and x = − b , which can be visualized as the points where the parabola intersects the x-axis. These concepts are fundamental in modeling and analyzing various phenomena.
