Domain and range of exponential and logarithmic functions

Learning Objectives Determine the domain and range of a base exponential function, f(x) = b^x. Determine the domain and range of a base logarithmic function, f(x) = log_b(x). Analyze transformations to find the domain and range of complex exponential functions of the form f(x) = a * b^(k(x-d)) + c. Analyze transformations to find the domain and range of complex logarithmic functions of the form f(x) = a * log_b(k(x-d)) + c. Identify the equation of the horizontal asymptote for an exponential function and explain its relationship to the range. Identify the equation of the vertical asymptote for a logarithmic function and explain its relationship to the domain. Explain how the inverse relationship between exponential and logarithmic functions causes their domains and ranges to...

TermDefinitionExample DomainThe set of all possible input values (often x-values) for which a function is defined.For f(x) = log(x), the domain is (0, ∞) because the logarithm of a non-positive number is undefined. RangeThe set of all possible output values (often y-values) that a function can produce from its domain.For f(x) = 2^x, the range is (0, ∞) because no matter the value of x, 2^x is always a positive number. Exponential Function (General Form)A function of the form f(x) = a * b^(k(x-d)) + c, where b > 0 and b ≠ 1. Its graph features a horizontal asymptote.f(x) = 3 * 2^(x-1) + 5 Logarithmic Function (General Form)A function of the form f(x) = a * log_b(k(x-d)) + c, where b > 0 and b ≠ 1. It is the inverse of the exponential function and its graph features a vertical asympto...

Domain and Range of Transformed Exponential Functions For f(x) = a * b^(k(x-d)) + c: Domain: (-∞, ∞) Range: (c, ∞) if a > 0, or (-∞, c) if a < 0. Horizontal Asymptote: y = c The domain of any exponential function is all real numbers. The range is determined entirely by the vertical shift (c) and whether the function is reflected across the horizontal asymptote (sign of a). Domain and Range of Transformed Logarithmic Functions For f(x) = a * log_b(k(x-d)) + c: Domain: Solve the inequality k(x-d) > 0 for x. Range: (-∞, ∞) Vertical Asymptote: x = d The range of any logarithmic function is all real numbers. The domain is restricted to values that make the argument of the logarithm positive. The vertical asymptote occurs at the boundary of this domain.