Consider the graph below of the function f.

The domain of f is _______.

Question

Consider the graph below of the function f.

The domain of f is _______.

BlackTom AI · Accepted Answer

To begin, I will restate the question and list the answer choices clearly so we can reference them as we analyze each option.
Question: Consider the graph below of the function f. The domain of f is _______.
Answer choices:
- [-1,+∞)
- (-∞,-3] ∪ [3,+∞)
- [-3,3]
- (-∞,-1] ∪ [-1,+∞)

Now, let's evaluate each option one by one.

Option 1: [-1, +∞). This would mean the function is defined for all x values from -1 onward to positive infinity, but undefined for x < -1. Observing the graph, there are valid x-values extending far to the left (negative side) where the function is defined, so this option is inconsistent with the graph.

Option 2: (-∞, -3] ∪ [3, +∞). This option describes a domain consisting of all x-values up to and including -3, and from 3 onward to the right, with a gap between -3 and 3 where the function is not defined. If the graph shows that the function has holes or is not defined between -3 and 3, and is defined outside this interval, this option would align with that pattern.

Option 3: [-3, 3]. This would indicate the function is only defined on the finite interval from -3 to 3, including both endpoints. If the graph demonstrates the function exists beyond x = -3 and x = 3, then this option would be incorrect because the domain would be too restricted.

Option 4: (-∞, -1] ∪ [-1, +∞). This is a bizarre way of writing the entire real line with a potential duplication at -1, effectively suggesting the function is defined for all real x. If the graph clearly shows a gap somewhere (for example, between some symmetric positive value and negative value), this option would be inaccurate because it implies no missing values in the domain.

Next, comparing these interpretations to the visual cues in the graph: the function appears to be defined for large negative x values up to -3 inclusive, and again for x values from 3 inclusive onward to the right, with a clear gap between -3 and 3 where the function is not defined. This pattern matches Option 2 precisely and contradicts Option 1 (which starts at -1), Option 3 (which confines to [-3,3]), and Option 4 (which would cover all real numbers or misrepresent the domain).

In summary, Option 2 reflects a domain that excludes the interval (-3, 3) and includes the endpoints at -3 and 3, matching the left branch and right branch of the graph with a gap in between. The remaining options fail to align with the observed domain coverage shown by the graph, either by extending too far left, restricting too much, or implying the entire real line is included.

Consider the graph below of the function f. The domain of f is _______.单项选择题

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