Both of these functions grow as x gets larger and larger. Which function eventually exceeds the other? f(x)=5x^2-8 g(x)=3^x

To determine which function eventually exceeds the other as gets larger, we need to analyze the growth rates of the two functions: ### Analysis of Growth Rates1. **Quadratic Function \( f(x) = 5x^2 - 8 \)**: - The term dominates the behavior of the function for large values of . - The growth rate of a quadratic function is proportional to .2. **Exponential Function \( g(x) = 3^x \)**: - The term grows exponentially. - The growth rate of an exponential function is proportional to the function's current value, meaning it increases faster and faster as increases.### Comparison of Growth RatesFor large values of , exponential growth outpaces quadratic growth. This can be understood by comparing the rates of increase:- For sufficiently large, will grow much faster than .To provide a more concrete comparison, let's calculate the values of \( f(x) \) and \( g(x) \) for a few large values of :- For : - For : From these calculations, it is evident that \( g(x) = 3^x \) grows much faster than \( f(x) = 5x^2 - 8 \) for large values of .### ConclusionThe exponential function \( g(x) = 3^x \) will eventually exceed the quadratic function \( f(x) = 5x^2 - 8 \) as gets larger.