Domain and range of exponential functions

Learning Objectives Define domain and range in the context of exponential functions. Identify the domain of any exponential function of the form f(x) = a * b^(x-h) + k. Identify the horizontal asymptote of a transformed exponential function. Determine the range of a transformed exponential function by analyzing its vertical shift (k) and reflection (a). Express the domain and range of exponential functions using both interval notation and set-builder notation. Explain how transformations, specifically vertical shifts, affect the range of an exponential function. 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, we'll explore the 'inputs' (...

TermDefinitionExample Exponential FunctionA function where the variable is in the exponent, typically in the form f(x) = b^x, where the base 'b' is a positive constant other than 1.f(x) = 2^x is an exponential growth function. g(x) = (1/2)^x is an exponential decay function. DomainThe complete set of all possible input values (x-values) for which the function is defined.For the function f(x) = 3^x, you can plug in any real number for x (like -2, 0, 5.5), so the domain is all real numbers. RangeThe complete set of all possible output values (y-values) that a function can produce.For f(x) = 3^x, the output y is always a positive number (e.g., 3^-2 = 1/9, 3^0 = 1, 3^2 = 9). The range is all positive real numbers, y > 0. Horizontal AsymptoteA horizontal line that the graph of a f...

Domain of an Exponential Function For any exponential function f(x) = a * b^(x-h) + k, the domain is always the set of all real numbers. This is because you can substitute any real number for x in the exponent. There are no restrictions like division by zero or square roots of negative numbers. In set-builder notation: {x | x ∈ ℝ}. In interval notation: (-∞, ∞). Range of an Exponential Function For f(x) = a * b^(x-h) + k, the horizontal asymptote is y = k. The range depends on the sign of 'a'. If 'a' is positive (a > 0), the graph opens upward from the asymptote, so the range is y > k. If 'a' is negative (a < 0), the graph is reflected and opens downward from the asymptote, so the range is y < k.