Question

Which function grows at the fastest rate for increasing values of x?

1. \( g(x) = 15x + 6 \)
2. \( f(x) = 4 \cdot 2^x \)
3. \( h(x) = 9x^2 + 25 \)

Answer 1

To determine which function grows at the fastest rate for increasing values of x, we need to compare the growth rates of the three functions. The function f(x) = 4 · 2^x grows at the fastest rate for increasing values of x.

Step-by-step explanation:

To determine which function grows at the fastest rate for increasing values of x, we need to compare the growth rates of the three functions.

Let's analyze each function:

1. g(x) = 15x + 6
   This is a linear function with a constant growth rate of 15. The coefficient of x represents the growth rate, which remains the same for all values of x.
2. f(x) = 4 · 2^x
   This is an exponential function with a base of 2 and a growth rate that increases exponentially. As x increases, 2^x grows at an accelerating rate, resulting in a faster growth compared to a linear function.
3. h(x) = 9x^2 + 25
   This is a quadratic function with a growth rate that increases quadratically. As x increases, the growth rate increases faster than a linear function but slower than an exponential function.

Therefore, f(x) = 4 · 2^x grows at the fastest rate for increasing values of x.