Question
short define absolute function, step functions,discontinuous, continuous,vertex,range
Answers
GPT-4o mini
Sure! Here are short definitions for each term:
1. **Absolute Function**: A mathematical function that returns the non-negative value of its input, denoted as \( |x| \), where \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \).
2. **Step Functions**: Piecewise constant functions that jump from one value to another without taking on any intermediate values. They are often represented graphically as a series of horizontal segments.
3. **Discontinuous**: A property of a function where it is not continuous at one or more points in its domain. This typically means there are jumps, breaks, or holes in its graph.
4. **Continuous**: A property of a function where it is unbroken and has no gaps, jumps, or holes over its domain. Mathematically, for any point in its domain, the limit as the function approaches that point equals the function's value at that point.
5. **Vertex**: The highest or lowest point of a parabola (in the context of quadratic functions) or a point where lines meet. For quadratic functions, it is often given by the coordinates \((h, k)\) in the vertex form \(y = a(x - h)^2 + k\).
6. **Range**: The set of all possible output values (y-values) of a function. It describes the extent of the function's values based on its domain. For example, in the case of a quadratic function opening upwards, the range would start from the y-coordinate of the vertex to positive infinity.
1. **Absolute Function**: A mathematical function that returns the non-negative value of its input, denoted as \( |x| \), where \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \).
2. **Step Functions**: Piecewise constant functions that jump from one value to another without taking on any intermediate values. They are often represented graphically as a series of horizontal segments.
3. **Discontinuous**: A property of a function where it is not continuous at one or more points in its domain. This typically means there are jumps, breaks, or holes in its graph.
4. **Continuous**: A property of a function where it is unbroken and has no gaps, jumps, or holes over its domain. Mathematically, for any point in its domain, the limit as the function approaches that point equals the function's value at that point.
5. **Vertex**: The highest or lowest point of a parabola (in the context of quadratic functions) or a point where lines meet. For quadratic functions, it is often given by the coordinates \((h, k)\) in the vertex form \(y = a(x - h)^2 + k\).
6. **Range**: The set of all possible output values (y-values) of a function. It describes the extent of the function's values based on its domain. For example, in the case of a quadratic function opening upwards, the range would start from the y-coordinate of the vertex to positive infinity.