Answer:
AI-generated answer
To sketch the graph of the absolute value function ƒ(x) = |x + 3| - 1 and find its domain and range, we can follow these steps:
1. Graphing the function:
- Start by analyzing the expression inside the absolute value bars, which is x + 3. This means that when x is greater than or equal to -3, the value inside the absolute value will be positive, and when x is less than -3, it will be negative.
- Plot some key points on the graph to help visualize the function:
- When x = -3, the absolute value becomes |0| = 0, so the point (-3, 0) is on the graph.
- When x = 0, the absolute value becomes |3| = 3, so the point (-3, 2) is on the graph.
- When x = -6, the absolute value becomes |-3| = 3, so the point (-6, 2) is on the graph.
- Connect these points with a V-shaped graph, with the vertex at (-3, -1).
2. Determining the domain:
- The domain of a function is the set of all possible x-values for which the function is defined.
- Since the absolute value function is defined for all real numbers, the domain of ƒ(x) = |x + 3| - 1 is (-∞, ∞), meaning all real numbers.
3. Finding the range:
- The range of a function is the set of all possible y-values that the function can take.
- In this case, the minimum value of the function occurs at the vertex of the graph, which is (-3, -1). So, the range is (-∞, -1].
To summarize:
- The graph of ƒ(x) = |x + 3| - 1 is a V-shaped graph with the vertex at (-3, -1).
- The domain of the function is (-∞, ∞), which includes all real numbers.
- The range of the function is (-∞, -1], meaning all real numbers less than or equal to -1.
Explanation: