Both of these functions grow as x gets larger and larger. Which function eventually exceeds the other? f(x)=0.5x^2+x g(x)=0.6x+8

To determine which function eventually exceeds the other as x gets larger, we need to compare the growth rates of the two functions.<br /><br />The function $f(x)=0.5x^{2}+x$ is a quadratic function, which means its growth rate is proportional to the square of x. On the other hand, the function $g(x)=0.6x+8$ is a linear function, which means its growth rate is proportional to x.<br /><br />As x gets larger and larger, the quadratic function $f(x)$ will eventually exceed the linear function $g(x)$ because the growth rate of the quadratic function is faster than that of the linear function.<br /><br />Therefore, the function $f(x)=0.5x^{2}+x$ will eventually exceed the function $g(x)=0.6x+8$ as x gets larger and larger.