Question 4

The graphs will intersect only at the point (1,1).

The graphs of the functions f(x) = x and g(x) = 3x will intersect at the point (1,1), where both functions yield the same output for the input of 1.

A) The graphs will not intersect.

This statement is incorrect because the two graphs do intersect at a specific point. Both functions are linear and will cross each other at one point since they have different slopes.

B) The graphs will intersect only at the point (0,0).

While the point (0,0) is indeed a point where both graphs meet, it is not the only intersection point. Therefore, this statement is partially true but does not reflect the complete intersection details.

C) The graphs will intersect only at the point (0,1).

This statement is incorrect because the point (0,1) does not lie on either graph. For f(x) = x, the output at x = 0 is 0, and for g(x) = 3x, the output is also 0 at that point.

D) The graphs will intersect only at the point (1,1).

This statement is correct as both functions yield the value of 1 when x = 1. At this point, f(1) = 1 and g(1) = 3(1) = 3, confirming that they intersect only at (1,1).

E) The graphs will intersect only at the point (3,3).

This statement is incorrect because while (3,3) lies on the graph of f(x), it does not correspond to the output of g(x) = 3x at x = 3, which yields the point (3,9).

Conclusion

The correct answer is D, as it accurately identifies the only intersection point of the two functions f(x) = x and g(x) = 3x as (1,1). All other options either misidentify the intersection points or incorrectly state that intersections do not occur, emphasizing the importance of understanding the graphical relationships of linear equations.