As the value of x increases, which of the following functions would eventually exceed the others?
1) \( f(x) = 1000x \)
2) \( f(x) = 100x^2 \)
3) \( f(x) = 50x^3 \)
4) \( f(x) = 2^x \)
Answer (3)
For large values of x, the exponential function f(x)=2^x will exceed the growth rates of the linear, quadratic, and cubic functions due to its more rapid increase.
As the value of x increases, to determine which function will eventually exceed the others, we need to compare the growth rates of the given functions: f(x)=1000x, f(x)=100x², f(x)=50x³, and f(x)=2x. For large values of x , the exponential function f(x)=2x will exceed the linear, quadratic, and cubic functions because exponential growth is more rapid than polynomial growth. An example illustrating this concept is that in each subsequent time period, the exponential function's value will double, while polynomial functions' increased rates will be limited by their highest power.
As the value of x increases, the function that would eventually exceed the other three is f(x) = 2ˣ. While linear, quadratic, and cubic functions all grow at different rates, the exponential growth of the exponential function surpasses them all over time. For example, while f(x) = 1000x and f(x) = 100x² grow linearly and quadratically respectively, and f(x) = 50x ³ grows cubically, the growth rates of these polynomial functions are outstripped by the exponential function due to the nature of exponential growth, where the rate of increase becomes faster in relation to the size of the quantity itself. In essence, despite the large coefficients in the polynomial functions, the exponential growth dominates for large values of x.
As x increases, the exponential function f ( x ) = 2 x will eventually exceed the other functions f ( x ) = 1000 x , f ( x ) = 100 x 2 , and f ( x ) = 50 x 3 . This is due to the faster growth rate of exponential functions compared to polynomial functions. Thus, the chosen option is f ( x ) = 2 x .
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