How Well Do You Know Absolute Value & Piecewise Functions?

Absolute value and piecewise functions primarily fall under the field of mathematics. These concepts are fundamental in algebra and calculus, which are branches of mathematics. Absolute value functions involve finding the distance between a number and zero, while piecewise functions involve defining different functions for different intervals of the input. Both concepts are extensively studied and applied in mathematics.

Discontinuous piecewise functions can be identified by looking for breaks, holes, and jumps in the graph. These are points where the function is not continuous and there is a discontinuity in the graph. Smooth transitions are not present in these functions, and instead, there are disjointed lines used for representation. Therefore, breaks, holes, and jumps are the key indicators for identifying discontinuous piecewise functions.

A piecewise function is defined by different equations in different intervals or regions. At the least, a piecewise function requires two equations to define it. One equation will define the function in one interval and another equation will define it in a different interval. Therefore, the correct answer is 2.

The step function is often referred to as the "staircase function" because its graph resembles a series of connected stair steps. It is called so because the function jumps from one constant value to another at specific intervals, creating a step-like pattern. The term "staircase" accurately describes this behavior and is commonly used to describe the step function.

A piecewise defined function can have both continuous and discontinuous parts. This means that certain intervals of the function may exhibit continuity, while other intervals may have points of discontinuity. Therefore, the statement "Continuous and discontinuous" is a fact in piecewise defined functions.

When the graph opens downwards, it means that the vertex of the parabola is at its highest point. In this case, the range of the function will be less than or equal to the y-coordinate of the vertex. This is because the y-coordinate of the vertex represents the maximum value that the function can attain, and since the graph is opening downwards, all the y-values of the graph will be less than or equal to the y-coordinate of the vertex. Therefore, the range will be limited to values that are less than or equal to the y-coordinate of the vertex.

Piecewise functions are also commonly referred to as split-definition functions. This is because these functions are defined by different equations or expressions over different intervals or "pieces" of the function's domain. Each piece of the function has its own separate definition, hence the term "split-definition functions". The other options listed in the question, such as "split functions" and "divided functions", are not commonly used terms to describe piecewise functions.

When finding the range of an absolute value function, you look for the vertex. The vertex represents the lowest or highest point on the graph, depending on whether the absolute value function is concave up or concave down. In this case, the correct answer is "Vertex" because it correctly identifies that the vertex is important when determining the range of an absolute value function.

Piecewise functions do not have a specific parent function. A parent function is a basic function that can be modified through transformations to create different functions. However, piecewise functions are defined differently than regular functions and involve different rules and conditions for different intervals or pieces of the function. Therefore, they do not have a single parent function that can be modified.

A piecewise function is a function that is defined by different expressions or formulas for different intervals or "pieces" of its domain. The limitations of a piecewise function in terms of form refer to the types of functions that can be used to define each piece of the function. The correct answer, "Exponential only," means that the pieces of the function can only be defined using exponential functions. This means that linear functions or other forms, such as cubic functions, cannot be used to define the pieces of the function.