Math Problem Statement
Please answer the following questions
Solution
Let's analyze the given graph and answer each question in detail based on the function .
(a) The domain of
The domain of a function represents all the possible values of for which is defined. From the graph:
- The function exists from to with some open circles indicating points where the function is not defined at specific -values.
- The function is undefined at and (indicated by open circles at these points).
Thus, the domain of is:
(b) The range of
The range is the set of all possible -values the function takes. Based on the graph:
- The minimum -value is (reached at ).
- The maximum -value seems to be unbounded as the graph increases for .
- The function skips some values, like it doesn’t exist at (corresponding to ).
Thus, the range of is:
(c)
From the graph, at , the function has a solid point at .
(d)
At , there is an open circle at , indicating that the function is not defined at .
(e)
This represents . First, we need to find :
- From the graph, .
Next, we evaluate :
- From the graph, .
Thus:
(f) Find a nonzero value of for which is not defined.
The expression will be undefined when the denominator is zero, i.e., when:
From the graph, at .
Thus, the nonzero value of is:
However, strictly speaking, is the solution if non-zero wasn't specified.
Let me know if you want further details or have any questions! Here are 5 related questions for deeper understanding:
- How would the graph change if had a discontinuity at more points?
- What would the domain be if was defined at ?
- How do we calculate in general for composite functions?
- How does the range of a function influence the solutions of ?
- How can we interpret the undefined points in terms of limits?
Tip: Always check for open and closed points in graphs to identify where a function is defined or undefined.
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Math Problem Analysis
Mathematical Concepts
Functions
Domain and Range
Composite Functions
Undefined Expressions
Formulas
g(x) = 1/(1 + f(x))
(f ∘ f)(x) = f(f(x))
Theorems
-
Suitable Grade Level
Grades 9-12