Math Problem Statement

Find the domain of the function f ∘ g, where f(x) is represented by the dashed line and g(x) by the solid line in the graph.

Solution

The problem asks for the domain of the composition of two functions fgf \circ g, using the graphs of ff (dashed line) and gg (solid line).

Step-by-step approach:

  1. Definition of Composition: The composition of two functions fgf \circ g means that f(g(x))f(g(x)). For this to exist, the output of g(x)g(x) must lie in the domain of ff. Thus, we need to find where g(x)g(x) gives values that lie within the domain of ff.

  2. Domain of ff: From the graph, f(x)f(x) (the dashed line) is defined for x[0,3]x \in [0, 3] since it only exists in this interval.

  3. Range of gg: g(x)g(x) (the solid line) is defined for x[0,6]x \in [0, 6], and the range of g(x)g(x) is from 0 to 3, inclusive.

  4. Condition for the Composition fgf \circ g: For f(g(x))f(g(x)) to exist, the output of g(x)g(x) must be within the domain of ff, which is from 0 to 3.

    • g(x)g(x) takes values between 0 and 3 for x[0,6]x \in [0, 6].

Thus, the domain of the composition fgf \circ g is the same as the domain of g(x)g(x), which is [0,6][0, 6].

Final Answer:

The domain of fgf \circ g is [0,6][0, 6].

Would you like further details or have any questions about this?

Here are 5 related questions for further exploration:

  1. What is the range of the composition function fgf \circ g?
  2. How does the composition fgf \circ g behave graphically?
  3. How do we determine the domain of fgf \circ g if the functions ff and gg are given algebraically?
  4. What is the importance of the range of g(x)g(x) in the composition f(g(x))f(g(x))?
  5. How would the domain change if the graph of f(x)f(x) were different?

Tip: Always check the range of the inner function when determining the domain of a composition function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Domain and Range

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12