The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. Notice that the graph in Figure \(\PageIndex{4}\) passes through the initial points given in the problem, \((−2, 6)\) and \((2, 1)\). The graph is asymptotic to the x-axis as x approaches negative infinity, The graph increases without bound as x approaches positive infinity, The line in the graph above is asymptotic to the x-axis as x approaches positive infinity, The line increases without bound as x approaches negative infinity. A function which grows faster than a polynomial function is y = f(x) = ax, where a>1. Lily will need to invest \($13,801\) to have \($40,000\) in \(18\) years. Thus, \(j(x)=(−2)^x\) does not represent an exponential function because the base, \(−2\), is less than \(0\). The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: To the nearest dollar, how much will Lily need to invest in the account now? In certain functions, either the value of the function tends to infinity (or –infinity) for an input variable or the function tends to a constant value at an infinitely small (or large) value of the input variable. This concept helps to find the asymptotes of exponential functions, which is shown as below: The curve represents the general form of an exponential function. It can be seen that as the exponent increases, the curves get steeper and the rate of growth increases respectively. What is \(f(3)\)? Thus, the exponential function is f(t) = 80 e.4581 t and there will be 3,125 bacteria at 8:00 PM. In this exponential function, \(100\) represents the initial number of stores, \(0.50\) represents the growth rate (the number of stores is growing by 50% each year), and \(1+0.5=1.5\) represents the growth factor. Here, \(4^{-6}\) is the initial value \(a\) and \(64\) is the base \(b\). We want to find the initial investment, \(P\), needed so that the value of the account will be worth \($40,000\) in \(18\) years. The basic curve of an exponential function looks like the following: Introduction to asymptotes of exponential functions. 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