Domain and range of absolute value functions

Learning Objectives Define domain and range in the context of absolute value functions. Identify the vertex of an absolute value function from its equation in the form f(x) = a|x - h| + k. Determine the domain of any absolute value function. Determine the range of an absolute value function by analyzing its vertex and the direction of opening. Write the domain and range using both inequality notation and interval notation. Explain how the parameters 'a' and 'k' in the vertex form affect the range of the function. Imagine a bouncing ball. 🏀 Its height above the ground is always positive, no matter which direction it's moving. How can we describe all the possible heights it can reach? In this tutorial, we will explore the domain (all possible x-value...

TermDefinitionExample Absolute Value FunctionA function that contains an algebraic expression within absolute value symbols. Its graph is a V-shape.f(x) = |x| or g(x) = -2|x - 1| + 5 DomainThe set of all possible input values (x-values) for which the function is defined.For f(x) = |x|, you can plug in any real number for x, so the domain is all real numbers. RangeThe set of all possible output values (y-values) that the function can produce.For f(x) = |x|, the output is never negative, so the range is all non-negative numbers (y ≥ 0). VertexThe point where the absolute value graph changes direction. It is either the lowest point (minimum) or the highest point (maximum) of the graph.The vertex of f(x) = |x - 3| + 4 is at the point (3, 4). Interval NotationA way of writing subsets of real n...

Vertex Form of an Absolute Value Function f(x) = a|x - h| + k This is the standard form used to find the key features of the function. The vertex is at the point (h, k). The value 'a' determines the vertical stretch/compression and the direction of opening (up if a > 0, down if a < 0). Finding the Domain Domain: (-∞, ∞) or {x | x ∈ ℝ} For any absolute value function of the form f(x) = a|x - h| + k, the domain is always all real numbers. This is because there are no restrictions on the x-values you can substitute into the function. Finding the Range If a > 0, Range is [k, ∞). If a < 0, Range is (-∞, k]. The range depends entirely on the vertex's y-coordinate (k) and the direction of opening (determined by 'a'). If the V-shape opens...