Question

Two functions each have an initial value of 1. One function is linear and has a slope of 10 while the other function is exponential with a growth factor of 1.1.
At what x-value does the exponential function’s output exceed the linear?

Why does this happen?

Answer 1

Answer:

x=68.53

Step-by-step explanation:

The linear function has an initial value of 1, meaning the the y-intercept is 1.

The slope of the linear function is 10.

Its equation is

$y = 10x + 1$

The exponential function has equation of the form

$y = a( {1.1})^{x}$

Since, the graph goes through (0,1) we can find the value of 'a'

$1 = a( {1.1)}^{0}$

$1 =a (1)$

$a = 1$

Hence the exponential function has equation:

$y = {1.1}^{x}$

We graph the two functions;

The exponential function will overtake the linear graph at x= 68.53

Although the linear function was initially above the exponential, it is increasing arithmetically while the exponential is growing geometrically so it will eventually overtake the linear function.

In other words the exponential function is growing at a multiplicative rate while the linear function is growing at an additive rate.