Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = 9x + 7$
$g(x) = 4.9^x - 5$

Explanation:

Step1: Analyze the types of functions

$f(x) = 9x + 7$ is a linear function (degree 1 polynomial), and $g(x)=4.9^{x}-5$ is an exponential function with base $4.9>1$.

Step2: Recall the growth rates of functions

Linear functions have a constant rate of change (slope), while exponential functions with base $> 1$ grow at an increasing rate. As $x$ becomes very large, exponential functions with base greater than 1 will eventually outpace linear functions.

Step3: Compare the two functions for large $x$

For large values of $x$, the term $4.9^{x}$ in $g(x)$ will grow much faster than the linear term $9x$ in $f(x)$. Even though $g(x)$ has a subtraction of 5 and $f(x)$ has an addition of 7, the exponential growth of $4.9^{x}$ will dominate as $x$ increases without bound. So, $g(x)$ will eventually exceed $f(x)$.

Answer:

The function $g(x)=4.9^{x}-5$ eventually exceeds $f(x) = 9x + 7$.