Evaluate an exponential function

Learning Objectives Define an exponential function and identify its components (base, exponent, initial value). Substitute a given input value for the variable in an exponential function. Correctly apply the order of operations when evaluating exponential functions. Evaluate exponential functions for positive integer, negative integer, and zero exponents. Use a calculator to approximate the value of an exponential function for non-integer exponents. Interpret the result of evaluating an exponential function within a real-world context. Ever wonder how a single social media post can get millions of views in just a few hours? 🤯 That's the power of exponential growth! In this tutorial, you will learn the fundamental skill of evaluating exponential functions. This means p...

TermDefinitionExample Exponential FunctionA function where the input variable (x) appears as an exponent. The general form is f(x) = a ⋅ b^x, where 'a' is the initial value and 'b' is the constant base.f(x) = 3 ⋅ 2^x is an exponential function. Here, the initial value is 3 and the base is 2. Base (b)The number that is repeatedly multiplied by itself. In f(x) = a ⋅ b^x, 'b' is the base. It must be a positive number and not equal to 1.In the function g(t) = 10(1.5)^t, the base is 1.5. Exponent (x)The variable that indicates how many times the base is to be multiplied by itself.In the function h(x) = 5^x, 'x' is the exponent. If we evaluate h(3), the exponent is 3, meaning 5 ⋅ 5 ⋅ 5. Initial Value (a)The starting amount of the function, or the value of...

The Exponential Function Form f(x) = a \cdot b^x This is the standard form. To evaluate, substitute the given value for 'x', calculate the power b^x first, and then multiply by 'a'. Zero Exponent Rule b^0 = 1 \quad (for\ b \neq 0) Any non-zero base raised to the power of zero is equal to 1. This is useful for finding the initial value of a function, f(0) = a ⋅ b^0 = a ⋅ 1 = a. Negative Exponent Rule b^{-x} = \frac{1}{b^x} A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. This rule is essential for evaluating functions for negative inputs.