Determining the range of a decreasing function requires understanding its behavior over its entire domain. Here's how you can approach it:
1. Understand Decreasing Functions
A function f(x) is decreasing over an interval if, for any two points x₁ and x₂ in that interval, where x₁ < x₂, then f(x₁) > f(x₂). In simpler terms, as x increases, y (or f(x)) decreases.
2. Determine the Domain
First, identify the domain of the function. The domain is the set of all possible x-values for which the function is defined. This is crucial because the range depends on the domain.
3. Find Critical Points (Where the Derivative is Zero or Undefined)
While not directly used to find the range, knowing where a function changes from increasing to decreasing (or vice-versa) is helpful.
- Calculate the derivative: Find f'(x), the derivative of the function.
- Set the derivative to zero: Solve f'(x) = 0 to find critical points.
- Identify points where the derivative is undefined: These are also critical points.
4. Identify Intervals of Decrease
- Test intervals: Choose test values within the intervals defined by the critical points and the boundaries of the domain. Plug these test values into the derivative f'(x).
- Determine where f'(x) < 0: If f'(x) is negative in an interval, the function is decreasing over that interval.
5. Determine the Range Over the Decreasing Interval(s)
Once you've identified the interval(s) where the function is decreasing, you can determine the range within those intervals. This involves finding the minimum and maximum y-values (or limits as x approaches endpoints of the interval).
- Evaluate the function at the endpoints of the decreasing interval(s): Find f(a) and f(b), where a and b are the endpoints of the interval(s) where the function is decreasing. Keep in mind that one or both of a and b may be infinite.
- Consider limits as x approaches infinity (if applicable): If the interval extends to infinity, evaluate the limit of f(x) as x approaches positive or negative infinity.
- Determine the range based on the function's behavior:
- If the function is decreasing from a to b, and both are finite: The range on that interval is [f(b), f(a)]. Remember the order is reversed because the function is decreasing!
- If the function is decreasing as x approaches infinity: The range may be bounded by a horizontal asymptote or approach infinity itself.
- Consider any local minima or maxima within the interval. Even if the function is decreasing, if there's a minimum or maximum within the interval, you will need to account for it.
Example:
Let's say f(x) = 1/x for x > 0.
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Domain: x > 0
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Derivative: f'(x) = -1/x²
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Interval of Decrease: f'(x) is always negative for x > 0. Therefore, the function is decreasing on the entire domain (0, ∞).
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Range:
- As x approaches 0 from the right, f(x) approaches ∞.
- As x approaches ∞, f(x) approaches 0.
Therefore, the range of f(x) on the interval (0, ∞) is (0, ∞).
Table Summarizing the Steps:
| Step | Description |
|---|---|
| 1. Domain Identification | Find all valid x values for the function. |
| 2. Derivative Calculation | Calculate f'(x). |
| 3. Identify Decreasing Intervals | Solve f'(x) < 0 to find intervals where the function is decreasing. |
| 4. Evaluate Endpoints/Limits | Find function values at the interval endpoints, or the limits as x approaches infinite endpoints. |
| 5. Determine the Range | Based on the decreasing behavior and endpoints, determine the corresponding y-values that make up the range. |
In conclusion, finding the range of a decreasing function involves analyzing its behavior over its domain, particularly the intervals where its derivative is negative, and carefully considering the function's values at the boundaries of those intervals.
