Final answer:
For increasing values of x, the function h(x) = 2^x, which is an exponential function, grows at the fastest rate when compared to the quadratic function f(x) = 8x² - 3x. option b is the correct answer.
Step-by-step explanation:
The question relates to comparing the growth rates of two different functions: the exponential function h(x) = 2^x and the quadratic function f(x) = 8x² - 3x. For increasing values of x, the function that grows at the fastest rate is the exponential function, which by nature increases exponentially. To illustrate, with an exponential growth rate, after 5 doubling intervals, the value of h(x) will be 2µ = 32. In contrast, the quadratic function grows at a slower rate compared to exponential growth. Therefore, for very large values of x, h(x) = 2^x will outpace the growth rate of f(x) = 8x² - 3x.
The function that grows at the fastest rate for increasing values of x is h(x) = 2^x.
Exponential growth is characterized by the scale increasing as a power of some base raised to the time interval. In this case, the base is 2 and the function will double for each increment of x. For example, if x = 1, h(x) = 2^1 = 2, and if x = 2, h(x) = 2^2 = 4. The function grows exponentially as x increases.
In contrast, the function f(x) = 8x² - 3x does not exhibit exponential growth. It is a quadratic function with a maximum rate of increase at a specific value of x, but it does not grow exponentially for increasing values of x.