Domain and range

Learning Objectives Define domain and range in the context of functions. Determine the domain and range of a function from its graph. Algebraically find the domain of polynomial, rational, and radical functions. Algebraically find the domain of logarithmic functions. Express domain and range accurately using interval notation. Analyze and state the practical domain and range for functions modeling real-world scenarios. If a vending machine is a function, what happens when you press a button for an item that's sold out? 🤔 Understanding a function's limits is key! This tutorial explores the fundamental concepts of domain and range, which are the sets of all possible inputs and outputs for a function. Mastering this topic is crucial for understanding the behavior, l...

TermDefinitionExample FunctionA mathematical relation where each input value (from the domain) is paired with exactly one output value (from the range).f(x) = x^2 is a function because every input x has only one output, x^2. For example, f(3) = 9. DomainThe complete set of all possible input values (often the 'x' values) for which a function is defined.For the function f(x) = √x, the domain is [0, ∞) because we cannot take the square root of a negative number in the real number system. RangeThe complete set of all possible output values (often the 'y' or 'f(x)' values) that a function can produce from the values in its domain.For the function f(x) = x^2, the range is [0, ∞) because squaring any real number results in a non-negative output. Interval NotationA...

Domain of a Rational Function For a function f(x) = g(x) / h(x), the domain consists of all real numbers x except those for which the denominator h(x) = 0. To find the domain, set the denominator equal to zero and solve for x. These solutions are the values that must be excluded from the domain. Domain of a Radical Function (Even Index) For a function f(x) = √[n]{g(x)} where n is an even integer (like 2, 4, ...), the domain consists of all real numbers x for which the radicand g(x) ≥ 0. To find the domain, set the expression inside the radical to be greater than or equal to zero and solve the inequality. Domain of a Logarithmic Function For a function f(x) = log_b(g(x)), the domain consists of all real numbers x for which the argument g(x) > 0. To find the domain,...