Evaluate exponential functions

Learning Objectives Evaluate an exponential function for a given integer value. Evaluate an exponential function for a given rational (fractional) value without a calculator. Use a calculator to approximate the value of an exponential function for an irrational input. Evaluate the natural exponential function, f(x) = e^x, for various inputs using a calculator. Substitute a given value into an exponential model to determine an output in a real-world context. Correctly apply the order of operations when evaluating exponential functions of the form f(x) = ab^x. Ever wonder how a single social media post can go viral, reaching millions of people in just a few hours? 📈 That's the power of exponential growth! In this tutorial, you will learn the fundamental skill of evaluat...

TermDefinitionExample Exponential FunctionA function of the form f(x) = ab^x, where 'a' is the non-zero initial value, 'b' is the base (b > 0 and b ≠ 1), and 'x' is the exponent.f(x) = 3(2)^x is an exponential function with an initial value of 3 and a base of 2. Base (b)The constant value that is raised to the power of the variable exponent. It determines the rate of growth (if b > 1) or decay (if 0 < b < 1).In the function g(t) = 10(0.5)^t, the base is 0.5, indicating exponential decay. Initial Value (a)The output of the function when the input (x) is zero. It represents the starting amount in many real-world models.For P(t) = 500(1.05)^t, the initial value is 500. This could represent an initial investment of $500. Rational ExponentAn exponent t...

The General Exponential Function Form f(x) = ab^x To evaluate, substitute the given value for 'x' into the exponent. Always calculate the power (b^x) first before multiplying by the coefficient 'a', following the order of operations (PEMDAS/BODMAS). Rational Exponent Rule b^{m/n} = (\sqrt[n]{b})^m = \sqrt[n]{b^m} Use this rule to evaluate exponential functions for fractional inputs. It's often easier to take the n-th root of the base 'b' first, then raise the result to the m-th power. Negative Exponent Rule b^{-x} = \frac{1}{b^x} If the exponent is negative, take the reciprocal of the base raised to the positive exponent. This is a key prerequisite for evaluating functions at negative inputs.