To determine which of the given functions has the fastest rate of growth as \( x \) approaches infinity, we can analyze each function's behavior as \( x \) increases:
A: \( y = 25x^2 \)
This is a polynomial function of degree 2. As \( x \) grows large, this function will grow at a rate proportional to \( x^2 \).
B: \( y = 1 - (1.8)^x \)
As \( x \) increases, \( (1.8)^x \) will grow without bound, meaning \( y \) will approach negative infinity. Hence, this function does not exhibit growth in the sense we are considering.
C: \( y = 3(2.1)^x \)
This is an exponential function where the base \( 2.1 > 1 \). Exponential functions grow significantly faster than polynomial functions as \( x \) becomes very large.
D: \( y = 10(0.8)^x \)
Here, \( (0.8)^x \) decreases towards zero as \( x \) increases. Thus, this function approaches zero and does not exhibit growth.
Now comparing the growth rates:
- A grows polynomially as \( x^2 \).
- B decreases towards negative infinity, which isn’t growth.
- C grows exponentially, as \( 3(2.1)^x \).
- D decreases towards zero.
Thus, the function that has the fastest rate of growth as \( x \) approaches infinity is:
C: \( y = 3(2.1)^x \).
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