Math Problem Statement

Find the domain and range of the composition g ∘ f. Write the answers in set notation.

Solution

In this problem, you are asked to find the domain and range of the composition of functions gfg \circ f, given the mappings of ff and gg between their respective domains and ranges.

Step 1: Analyze Function ff

From the left diagram:

  • Domain of ff: {4, 5, 6, 8}
  • Range of ff: {0, 1, 3, 5}

The mappings of ff are:

  • f(4)=0f(4) = 0
  • f(5)=1f(5) = 1
  • f(6)=3f(6) = 3
  • f(8)=5f(8) = 5

Step 2: Analyze Function gg

From the right diagram:

  • Domain of gg: {0, 1, 3, 5} (which is the same as the range of ff)
  • Range of gg: {2, 4, 6, 7}

The mappings of gg are:

  • g(0)=4g(0) = 4
  • g(1)=2g(1) = 2
  • g(3)=7g(3) = 7
  • g(5)=6g(5) = 6

Step 3: Composition gfg \circ f

The composition gfg \circ f means applying ff first, and then applying gg to the result. Let's go step by step for each element in the domain of ff:

  • (gf)(4)=g(f(4))=g(0)=4(g \circ f)(4) = g(f(4)) = g(0) = 4
  • (gf)(5)=g(f(5))=g(1)=2(g \circ f)(5) = g(f(5)) = g(1) = 2
  • (gf)(6)=g(f(6))=g(3)=7(g \circ f)(6) = g(f(6)) = g(3) = 7
  • (gf)(8)=g(f(8))=g(5)=6(g \circ f)(8) = g(f(8)) = g(5) = 6

Step 4: Find Domain and Range of gfg \circ f

  • Domain of gfg \circ f: This is the domain of ff, which is {4, 5, 6, 8}.
  • Range of gfg \circ f: The outputs of the composition gfg \circ f are {4, 2, 7, 6}.

Final Answer:

(a) Domain of gfg \circ f: {4, 5, 6, 8}

(b) Range of gfg \circ f: {2, 4, 6, 7}

Would you like any more details or clarifications on these concepts?


Here are 5 related questions for further practice:

  1. How do you find the composition of two functions algebraically, without mappings?
  2. What is the domain of a composition of functions if one function is undefined for certain values in its domain?
  3. Can the composition of two functions ever result in a function with a different domain than the original functions?
  4. How does the range of a function affect the composition with another function?
  5. Can functions be composed in the reverse order, i.e., fgf \circ g? How would the process differ?

Tip: Always check if the range of the first function matches the domain of the second function when composing two functions.

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Math Problem Analysis

Mathematical Concepts

Functions
Composition of Functions
Set Notation

Formulas

g ∘ f(x) = g(f(x))

Theorems

Composition of Functions Theorem

Suitable Grade Level

Grades 10-12