Each of these functions grows as x gets larger and larger. Which function eventually exceeds the others? f(x)=3^x g(x)=5x^2+x h(x)=8x+5

To determine which function eventually exceeds the others as x gets larger and larger, we need to analyze the growth rates of each function.<br /><br />Let's compare the functions:<br /><br />1. $f(x) = 3^x$<br />2. $g(x) = 5x^2 + x$<br />3. $h(x) = 8x + 5$<br /><br />As x gets larger and larger, the function that grows the fastest will eventually exceed the others.<br /><br />Comparing the growth rates of the functions:<br /><br />- $f(x) = 3^x$ is an exponential function, which grows faster than any polynomial function.<br />- $g(x) = 5x^2 + x$ is a quadratic function, which grows faster than any linear function.<br />- $h(x) = 8x + 5$ is a linear function, which grows slower than both the exponential and quadratic functions.<br /><br />Therefore, the function that eventually exceeds the others as x gets larger and larger is $f(x) = 3^x$.