Question 35

The graphs will not intersect.

The graphs of the functions f(x) = x and g(x) = 3^x do not intersect at any point in the standard (x, y) coordinate plane. This is due to the fact that as x increases, the exponential function g(x) grows much faster than the linear function f(x).

A) The graphs will not intersect.

This statement is true. The linear function f(x) = x increases at a constant rate, whereas the exponential function g(x) = 3^x increases rapidly for positive values of x. Hence, there are no points where the two graphs meet.

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

This statement is incorrect. While both functions equal zero at x = 0, the function g(x) = 3^x is always positive for any x greater than 0. Therefore, the graphs do not intersect at any point other than the origin, but they do not intersect anywhere else.

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

This statement is also incorrect. At x = 0, f(0) = 0 and g(0) = 3^0 = 1, indicating that the graphs do not intersect at this point.

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

This statement is incorrect as well. At x = 1, f(1) = 1 and g(1) = 3^1 = 3, showing that the graphs do not intersect at this point.

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

This statement is incorrect. At x = 3, f(3) = 3 and g(3) = 3^3 = 27, indicating that the graphs do not intersect here either.

Conclusion

In summary, the correct answer is that the graphs will not intersect, as the exponential function g(x) = 3^x will always be greater than the linear function f(x) = x for all positive x-values. All other options incorrectly assert points of intersection that do not occur based on the behavior of the two functions.