Math Problem Statement
Solve c
Solution
To solve part (c) of the problem, we are tasked with finding the domain and range of . Here’s a detailed breakdown:
1. Understanding the Problem
We are given . To determine the domain and range of , let’s analyze:
- Domain: The values of for which is defined.
- Range: The values of as varies over its domain.
The graph of is defined over as given.
2. Domain of
- The term means that the graph of is shifted 1 unit to the right.
- To ensure is defined, the argument must be within the domain of , which is .
Thus, solving for : Add 1 to all sides:
Therefore, the domain of is:
3. Range of
The range of depends on the range of , scaled by 2.
- From the graph of , the range of is .
- Since , the range of is obtained by multiplying the range of by 2.
Scaling the range by 2: Simplifying:
Thus, the range of is:
Final Answer:
- Domain of :
- Range of :
Let me know if you need further clarifications! Here are five related questions for practice:
- How does shifting affect its domain and range?
- If , how would affect the range of ?
- Determine the range of for the same function .
- What transformations are applied to to obtain ?
- If is reflected over the -axis, how would its range change?
Tip: Scaling a function by a constant multiplies the range values by .
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Math Problem Analysis
Mathematical Concepts
Functions
Domain and Range
Transformations of Functions
Formulas
Domain shift: If g(x) = f(x - h), domain shifts right by h.
Range scaling: If g(x) = kf(x), range is scaled by k.
Theorems
Function transformations and scaling
Suitable Grade Level
Grades 10-12
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