Suppose f is a function with exponential decay. Explain why the function g defined by g(x)=1/f(x) is a function with exponential growth.

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A general equation for exponential decay can be written f(x) = A e^(-kx), where A and k are positive contants.

If f(x) = A e^(-kx), then
1/f(x) = (1/A)e^kx ,

which is an equation for exponential growth.

Describe the important features of exponential decay.

To understand why the function g defined as g(x) = 1/f(x) exhibits exponential growth, we need to dive into the properties of exponential functions and how they relate to one another when we perform certain operations.

First, let's consider the function f(x) with exponential decay. Exponential decay means that as x increases, the value of f(x) decreases exponentially. Mathematically, this can be represented as f(x) = a * e^(-bx), where a and b are constants, and e is Euler's number (approximately 2.718).

Now, let's examine the function g(x) = 1/f(x), which is defined by taking the inverse of f(x). By substituting f(x) into the definition of g(x), we have g(x) = 1 / (a * e^(-bx)).

To comprehend why g(x) exhibits exponential growth, let's simplify the expression by multiplying the numerator and denominator of 1 by e^(bx):

g(x) = e^(bx) / (a * e^(-bx)).

Now, we can rewrite the expression as:

g(x) = e^(bx + bx) / a.

Simplifying further, we get:

g(x) = e^(2bx) / a.

Since e^(2bx) is an exponential function with a positive exponent, it will grow rapidly as the value of x increases. Therefore, g(x) also exhibits exponential growth.

In summary, when you take the inverse of a function with exponential decay, the resulting function will have exponential growth. The key here is understanding the properties of exponential functions and how they relate to each other when we perform operations such as taking the inverse.