Math Problem Statement
The implied domain of the function f is the set of x-values such that −1≤x≤4. The range of f is the set of y-values such that 0≤y≤7.
What is the closed-form expression of a function which has the stated domain and range? To answer this question, play around with inverse trigonometric functions and transformations.
Solution
To construct a function with the specified domain and range , we can make use of inverse trigonometric functions and transformations.
Step 1: Consider the base function
We start by using the inverse sine function , which naturally has a domain of and a range of .
Step 2: Adjust the domain
We need to adjust the domain to . We do this by transforming so that the resulting function's argument fits within the inverse sine's domain.
Define: This transformation maps from the interval into the interval .
Step 3: Adjust the range
Now, we need to scale and shift the function so that its range becomes .
The range of is . To adjust this range to , we perform a linear transformation:
Step 4: Simplify the expression
The final expression for simplifies to:
Verification:
- Domain: The function is defined for , matching the given domain.
- Range:
- When , , so . The smallest value can achieve is .
- When , , so . The largest value can achieve is .
This confirms that the function satisfies the given domain and range conditions.
Would you like more details on any part of this process?
Follow-up Questions:
- How would the function change if the domain was instead?
- What happens to the function's range if the transformation applied to the sine inverse is different?
- How would you approach creating a similar function using the inverse cosine function instead?
- Can this method be extended to functions with different ranges, such as ?
- What would the graph of this function look like over the specified domain?
- How can we derive the function if the desired range was ?
- Is there a way to construct the function without using trigonometric transformations?
- How would the derivative of this function behave within the given domain?
Tip:
When adjusting the range of a function, remember to apply both scaling and shifting transformations systematically to ensure the entire desired range is covered.
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Math Problem Analysis
Mathematical Concepts
Inverse trigonometric functions
Transformations of functions
Formulas
Inverse sine function: \( \sin^{-1}(x) \)
Theorems
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Suitable Grade Level
Advanced High School or College
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