Math Problem Statement
Solution
Let's work through this question step by step:
The function is one-to-one.
(a) Finding the inverse of
- Start with .
- Solve for in terms of :
- Swap and to find the inverse function:
So, the inverse function is .
(b) Domain and range of and
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Domain of : Since is a cubic function, it is defined for all real numbers . Therefore:
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Range of : A cubic function is surjective (covers all real numbers). Therefore:
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Domain of : The domain of is the range of , which is:
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Range of : The range of is the domain of , which is:
(c) Graph , , and on the same coordinate axes
To graph these:
- is a cubic function shifted down by 5.
- is the reflection of over .
- is the line of symmetry.
Would you like me to sketch this graph for you?
Questions to Expand Learning:
- What are the key properties of one-to-one functions, and why are they necessary for finding an inverse?
- How would the inverse function change if instead of ?
- How can we verify if two functions are inverses of each other?
- What happens to the domain and range when you apply transformations to a cubic function, such as shifting or reflecting?
- Why is the line of symmetry for inverse functions?
Tip:
When solving for an inverse function, always verify your solution by checking if and .
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Domain and Range
Graphing Functions
One-to-One Functions
Formulas
Inverse function: y = f^{-1}(x)
Cubic root: x = \sqrt[3]{y}
Theorems
A one-to-one function has an inverse
Inverse functions reflect over y = x
Suitable Grade Level
Grades 10-12