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 \( x \) gets larger, we need to analyze the growth rates of the two functions:<br /><br />\[ f(x) = 5x^2 - 8 \]<br />\[ g(x) = 3^x \]<br /><br />### Analysis of Growth Rates<br /><br />1. **Quadratic Function \( f(x) = 5x^2 - 8 \)**:<br /> - The term \( 5x^2 \) dominates the behavior of the function for large values of \( x \).<br /> - The growth rate of a quadratic function is proportional to \( x^2 \).<br /><br />2. **Exponential Function \( g(x) = 3^x \)**:<br /> - The term \( 3^x \) grows exponentially.<br /> - The growth rate of an exponential function is proportional to the function's current value, meaning it increases faster and faster as \( x \) increases.<br /><br />### Comparison of Growth Rates<br /><br />For large values of \( x \), exponential growth outpaces quadratic growth. This can be understood by comparing the rates of increase:<br /><br />- For \( x \) sufficiently large, \( 3^x \) will grow much faster than \( 5x^2 \).<br /><br />To provide a more concrete comparison, let's calculate the values of \( f(x) \) and \( g(x) \) for a few large values of \( x \):<br /><br />- For \( x = 10 \):<br /> \[ f(10) = 5(10)^2 - 8 = 5 \cdot 100 - 8 = 500 - 8 = 492 \]<br /> \[ g(10) = 3^{10} = 59049 \]<br /><br />- For \( x = 20 \):<br /> \[ f(20) = 5(20)^2 - 8 = 5 \cdot 400 - 8 = 2000 - 8 = 1992 \]<br /> \[ g(20) = 3^{20} = 3486784401 \]<br /><br />From these calculations, it is evident that \( g(x) = 3^x \) grows much faster than \( f(x) = 5x^2 - 8 \) for large values of \( x \).<br /><br />### Conclusion<br /><br />The exponential function \( g(x) = 3^x \) will eventually exceed the quadratic function \( f(x) = 5x^2 - 8 \) as \( x \) gets larger.