Which of the following statements is true about the graphs of f(x) = x and g(x) = 3x in the standard (x, y) coordinate plane?
The graphs will intersect only at the point (0,0).
Both functions, f(x) = x and g(x) = 3x, are linear equations in the standard (x, y) coordinate plane. They cross each other at the origin (0,0) and diverge from that point, confirming that their only intersection occurs at this coordinate.
This statement is incorrect because the two graphs are linear and will intersect at least at the origin (0,0). Since both functions pass through this point, it is impossible for them not to intersect at all.
While it is true that the graphs intersect at (0,0), this statement does not capture all intersection points. The two lines intersect at (0,0) and will diverge thereafter, making this statement misleading.
This choice is false because the point (0,1) is not on either graph. f(0) = 0 and g(0) = 0, indicating that both functions equal zero at x = 0, not one.
This statement is correct as both functions equal (1,1) when x = 1. Therefore, the graphs of f(x) = x and g(x) = 3x intersect at this point, alongside the origin (0,0).
Although both functions reach the point (3,3) when x = 3, this statement is incorrect as it suggests that this is the only intersection. They also intersect at (0,0) and (1,1), making it an incomplete assertion.
The graphs of f(x) = x and g(x) = 3x intersect at multiple points, specifically (0,0) and (1,1). However, the statement about them intersecting only at (1,1) is misleading, as their shared intersection also occurs at the origin. Therefore, the correct understanding of their intersection points is essential to accurately interpret linear relationships in coordinate geometry.
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