Math Problem Statement

Find the domain and range of the composition of functions g ∘ f based on the provided mapping diagram.

Solution

To solve this problem, we need to find the domain and range of the composition of two functions gfg \circ f, which is denoted as g(f(x))g(f(x)).

Step 1: Analyze the Domains and Ranges of ff and gg

From the diagrams provided:

  • The domain of ff is the set of all possible input values for ff, which appears to be {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}.
  • The range of ff (which becomes the input for gg in the composition gfg \circ f) is {5,6,7,8}\{5, 6, 7, 8\}.

For function gg:

  • The domain of gg is {5,6,7,8}\{5, 6, 7, 8\}, which matches the range of ff.
  • The range of gg is {6,9}\{6, 9\}.

Step 2: Determine the Domain and Range of gfg \circ f

For the composition gfg \circ f:

  • Domain of gfg \circ f: This will be the domain of ff, since f(x)f(x) must produce values in the range of ff that are within the domain of gg. Therefore, the domain of gfg \circ f is {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}.

  • Range of gfg \circ f: This will be the range of gg when applied to the range of ff. Observing the mappings, we see that:

    • f(1)=5g(5)=6f(1) = 5 \Rightarrow g(5) = 6
    • f(2)=6g(6)=9f(2) = 6 \Rightarrow g(6) = 9
    • f(3)=7g(7)=6f(3) = 7 \Rightarrow g(7) = 6
    • f(4)=8g(8)=9f(4) = 8 \Rightarrow g(8) = 9
    • f(5)=8g(8)=9f(5) = 8 \Rightarrow g(8) = 9
    • f(6)=8g(8)=9f(6) = 8 \Rightarrow g(8) = 9

    This results in a range for gfg \circ f of {6,9}\{6, 9\}.

Final Answer

  • (a) Domain of gfg \circ f: {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}
  • (b) Range of gfg \circ f: {6,9}\{6, 9\}

Would you like further clarification on this solution or more details on the composition concept?

Related Questions:

  1. What is the general rule for determining the domain of a composite function?
  2. How can you identify the range of a function from a mapping diagram?
  3. What would happen if there were values in the range of ff that weren’t in the domain of gg?
  4. How do you write compositions of functions in function notation?
  5. Can a composite function have a larger range than either of the original functions?

Tip:

When finding the range of a composition gfg \circ f, always check each output of ff to see how it maps through gg. This will ensure you capture all possible outcomes for the composite function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Composition of Functions
Domain and Range

Formulas

-

Theorems

-

Suitable Grade Level

Grade 11-12