Compare the growth rate of the functions f(x) = 4x and g(x) = the square root of the quantity of 16 raised to the x power plus 2 raised to the x power.
a. f(x) grows faster than g(x).
b. f(x) and g(x) grow at the same rate.
c. g(x) grows faster than f(x).
d. It cannot be determined.
Well, comparing growth rates can be a little tricky, but let's dive into it, shall we?
First, let's look at the functions themselves. The function f(x) = 4x is a simple linear function, where the growth is directly proportional to x. On the other hand, g(x) is a bit more interesting. We have the quantity of 16 raised to the x power plus 2 raised to the x power under the square root. This means that g(x) has an exponential component.
Now, let's compare the growth rates. As x increases, the function f(x) = 4x grows linearly. No matter how big x gets, the growth rate remains constant. On the other hand, g(x) has an exponential component, which means that as x increases, g(x) grows at an increasingly faster rate.
Based on that, we can conclude that g(x) grows faster than f(x). So the answer is c. g(x) grows faster than f(x).
Hope that sheds some light on the matter!
To compare the growth rates of the functions f(x) = 4x and g(x) = √(16^x + 2^x), we can analyze the behavior of the functions as x approaches infinity.
For f(x) = 4x, as x gets larger and approaches infinity, the function grows without bound. This means that f(x) grows exponentially, with a constant rate of growth.
For g(x) = √(16^x + 2^x), as x gets larger and approaches infinity, the dominant term in the expression is 16^x. The other term, 2^x, becomes negligible in comparison. In other words, g(x) can be approximated by √(16^x).
Now, let's compare the growth rate of f(x) and g(x):
As x approaches infinity, f(x) = 4x grows exponentially, meaning it grows faster than any power of x.
On the other hand, g(x) can be approximated as √(16^x), which grows at a polynomial rate.
Since f(x) grows exponentially and g(x) grows at a polynomial rate, we can conclude that f(x) grows faster than g(x). Therefore, the correct answer is:
a. f(x) grows faster than g(x).
To compare the growth rate of the functions f(x) = 4x and g(x) = sqrt(16^x + 2^x), we'll compare their rates by examining their behavior as x approaches infinity.
First, let's consider f(x) = 4x. As x increases, f(x) will grow exponentially because the base is a constant 4, and the exponent x determines how many times 4 is multiplied by itself. This means that f(x) will increase much faster as x gets larger.
Now let's analyze g(x) = sqrt(16^x + 2^x). We can rewrite this expression as g(x) = sqrt((16^x) + (2^x)). As x increases, both terms inside the square root will also increase. However, the dominant term is 16^x because its base (16) is larger than the base of 2^x (2). Therefore, g(x) is primarily determined by the growth rate of 16^x.
Considering the exponential nature of 16^x, g(x) will also grow exponentially but at a slower rate than f(x) because the base (16) of the exponent is larger than 4. Thus, as x approaches infinity, the growth rate of g(x) will be slower than that of f(x).
Based on this analysis, we can conclude that f(x) grows faster than g(x). Therefore, the correct option is a. f(x) grows faster than g(x).
exponentials eventually grow faster than any polynomials
16^x + 2^x grows faster than 16x^2
g = 16^x + 2^x = 2^(4x) + 2^x
g' = ln2 * 5*2^x
let F(x) = (4x)^2
F' = 32x
so, as soon as ln32 * 2^x > 32x g outstrips F
x > 8.32