over which interval does the growth rate of the exponential function exceed the growth rate of the linear…

Explanation:

Step1: Recall growth - rate concept

The growth - rate of a function is given by its derivative. For a linear function (y = mx + b), the derivative (y'=m) (constant). For an exponential function (y = a\cdot b^{x}), the derivative (y'=a\cdot b^{x}\ln(b)) which is non - constant. Geometrically, the growth - rate is the slope of the tangent line to the function's graph.

Step2: Analyze the graph

At the point of intersection ((1,2)) of the linear and exponential functions, before (x = 1), the linear function has a steeper slope (greater growth - rate). After (x = 1), the exponential function's slope (growth - rate) starts to increase more rapidly. As (x) increases further, the exponential function's growth - rate far exceeds that of the linear function.

Answer:

(x>1). Since the only option that satisfies this condition among the given choices is (x > 2), the answer is (x>2).