Math Problem Statement
Solution
Let's analyze and solve this exercise based on the image provided:
The problem is based on a function whose graph is given. Let's go through each question in detail.
1) Determine , the domain of definition of .
From observing the graph, it appears that the function is defined from to , inclusive. Thus, the domain of is:
2) Is continuous on ?
To determine continuity, we need to check if there are any breaks, jumps, or holes in the graph of over its domain . Based on the provided graph, appears to be a smooth, continuous curve from to , with no interruptions. Therefore, is continuous on .
3) Determine the values of on various intervals:
Let’s evaluate the ranges of over the specified intervals:
-
:
Observing the graph from to , the function starts at and increases to a maximum value at (approximately 4).
So, . -
:
This is the open interval, so it excludes the endpoints and . The range within this interval is similar, but strictly between 2 and 4.
Thus, . -
:
From to , the function decreases to a minimum at around (near ) and then increases again to around .
Hence, . -
:
For this interval, the function ranges from to a minimum near around .
Thus, . -
:
Since the function is only defined up to , we interpret as , which spans the full range of the function. The minimum value is around and the maximum around .
Therefore, .
Would you like further explanations on any part of this solution?
Here are five follow-up questions to deepen understanding:
- How would the domain change if the function graph extended beyond ?
- What if there were discontinuities in the graph? How would that affect the continuity on ?
- Can we determine the exact values at specific points (like ) from this graph?
- What kind of function (e.g., polynomial, exponential) does this graph resemble, based on its shape?
- How would you determine the derivative of if you only had the graph?
Tip: When analyzing function ranges over intervals, carefully check the graph for local minima and maxima within each interval.
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Math Problem Analysis
Mathematical Concepts
Functions
Continuity
Domain and Range
Formulas
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Theorems
Definition of function continuity
Domain and range of a function
Suitable Grade Level
High School (Grades 10-12)