Math Problem Statement
The domain of the function f(x) is open square bracket, minus, 8, comma, 1, close square bracket[−8,1] and the range is open square bracket, 1, comma, 17, close square bracket[1,17]. Using interval notation, find the domain and range of g, of, x, equals, f, left parenthesis, one quarter, x, right parenthesisg(x)=f( 4 1 x).
Solution
Given the function , we need to find the domain and range of based on the given domain and range of .
Domain of :
The domain of is , meaning that is only defined for values of in this interval. For , we need to figure out what values of will result in inputs to that lie within the interval .
- To find the domain of , solve the inequality:
- Multiply the entire inequality by 4 to eliminate the fraction:
Thus, the domain of is .
Range of :
The range of is , and since is simply a transformation of via scaling, it does not affect the range. Hence, the range of remains the same as that of , which is .
Conclusion:
- Domain of :
- Range of :
Would you like further details or have any questions?
Here are 5 related questions to expand on this:
- What happens to the domain if the function was instead of ?
- How does a horizontal compression or stretch affect the domain of a function?
- What if had a different range? How would it impact 's range?
- Can you visualize what transformations are occurring to the graph of to get ?
- How would the domain change if we had for any constant ?
Tip: When transforming functions, pay close attention to how scaling factors inside the function affect the domain, as they cause horizontal stretches or compressions.
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Math Problem Analysis
Mathematical Concepts
Functions
Transformations
Interval Notation
Formulas
Domain transformation: If g(x) = f(kx), solve for x in the interval of f(x)
Range remains unchanged for horizontal transformations
Theorems
Function Transformation Rules
Properties of Functions
Suitable Grade Level
Grades 10-12
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