Exponential functions are fundamental in modeling dynamic processes where the rate of change is proportional to the current value. Defined by f(x) = bx, where b is a positive constant not equal to one, they form the basis for describing processes of growth and decay depending on whether the base b is greater than or less than one.
Exponential models describe situations where change occurs at a rate proportional to the current amount. These include phenomena such as bacterial proliferation, radioactive substance decay, and depreciation of assets. The general form of an exponential model can be expressed as
Here, A0 is the initial amount, b is the base indicating the growth (b > 1) or decay (0 < b < 1) factor, and t represents time or another continuous variable.
Transformations such as translations, reflections, and stretches alter the graphical representation of exponential functions but preserve their fundamental shape and asymptotic behavior. Additionally, the One-to-One Property of exponential functions ensures that if bx = by, then x = y, allowing for solving equations involving exponents through base comparison.
An exponential function is defined by a positive base—not equal to one—raised to a real-number exponent.
Its general form is an initial value multiplied by the base raised to an exponent.
When the base is greater than one, the function models growth. When it is between zero and one, non-inclusively, it models decay.
For example, consider a drug that reaches an initial concentration of 30 micrograms per milliliter in the bloodstream after injection and retains 75 percent of its concentration each hour.
In this case, the exponential function is 30 multiplied by 0.75 raised to the exponent t, representing the hours passed.
After one hour, the concentration drops to 22.5 micrograms per milliliter. After two hours, it becomes 16.9 micrograms per milliliter, and the pattern continues.
On a graph, the curve falls rapidly at first, then gradually flattens, approaching a horizontal asymptote at zero.
Since time cannot be negative, the domain is the non-negative real numbers. Because the concentration never reaches zero, the range includes only positive values.
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