Question

NO LINKS!! Describe the domain and range (in BOTH interval and inequality notation) for each function shown part 1a​

Answer 1

`\textsf{Domain}: \quad (-\infty, 6) \quad -\infty < x < 6`

`\textsf{Range}: \quad (-\infty,6] \quad -\infty < y\leq 6`

Step-by-step explanation:

The domain of a function is the set of all possible input values (x-values).

The range of a function is the set of all possible output values (y-values).

An open circle indicates the value is not included in the interval.

A closed circle indicates the value is included in the interval.

An arrow show that the function continues indefinitely in that direction.

Interval notation

• ( or ) : Use parentheses to indicate that the endpoint is excluded.
• [ or ] : Use square brackets to indicate that the endpoint is included.

Inequality notation

• < means "less than".
• > means "more than".
• ≤ means "less than or equal to".
• ≥ means "more than or equal to".

From inspection of the given graph, the function is not continuous and so the domain is restricted.

There is an open circle at x = 6.

Therefore, the domain of the function is:

• Interval notation: (-∞, 6)
• Inequality notation: -∞ < x < 6

From inspection of the given graph, the maximum value of y is 6.

The function continues indefinitely to negative infinity.

Therefore, the range of the function is:

• Interval notation: (-∞, 6]
• Inequality notation: -∞ < y ≤ 6

Answer 2

Domain as an inequality:
`\boldsymbol{\text{x} < 6 \ \text{ or } \ -\infty < \text{x} < 6}`

Domain in interval notation:
`\boldsymbol{(-\infty, 6)}`

Range as an inequality:
`\boldsymbol{\text{y} \le 6 \ \text{ or } \ -\infty < \text{y} \le 6}`

Range in interval notation:
`\boldsymbol{(-\infty, 6]}`

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Step-by-step explanation:

The domain is the set of allowed x inputs. For this graph, the right-most point is when x = 6. This endpoint is not part of the domain due to the open hole. The graph goes forever to the left to indicate
`\text{x} < 6` but I think
`-\infty < \text{x} < 6` is far more descriptive.

The second format directly leads to the interval notation of
`(-\infty, 6)`

Always use parenthesis for either infinity. We use a parenthesis for the 6 to tell the reader not to include it as part of the domain.

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The range is the set of possible y outputs.

The highest y can get is y = 6

Therefore, y = 6 or y < 6

The range can be described as
`\text{y} \le 6 \ \text{ or } \ -\infty < \text{y} \le 6` where the second format is better suited to lead directly to the interval notation
`(-\infty, 6]`

Use a square bracket to include the 6 as part of the range. We don't have any open holes at the peak mountain point.