Math Problem Statement

Use the graphs to evaluate the following composite function expressions: f(g(5)), g(f(4)), f(f(2)), and g(g(1)).

Solution

Let's analyze the graphs and evaluate each of the expressions step-by-step.

  1. For f(g(5))f(g(5)):

    • First, find g(5)g(5) on the graph of g(x)g(x). When x=5x = 5, g(5)=4g(5) = 4.
    • Next, use f(4)f(4) on the graph of f(x)f(x). When x=4x = 4, f(4)=2f(4) = 2.
    • So, f(g(5))=f(4)=2f(g(5)) = f(4) = 2.
  2. For g(f(4))g(f(4)):

    • First, find f(4)f(4) on the graph of f(x)f(x). We already know from the previous step that f(4)=2f(4) = 2.
    • Next, use g(2)g(2) on the graph of g(x)g(x). When x=2x = 2, g(2)=1g(2) = 1.
    • Thus, g(f(4))=g(2)=1g(f(4)) = g(2) = 1.
  3. For f(f(2))f(f(2)):

    • First, find f(2)f(2) on the graph of f(x)f(x). When x=2x = 2, f(2)=5f(2) = 5.
    • Next, use f(5)f(5) on the graph of f(x)f(x). When x=5x = 5, f(5)=1f(5) = 1.
    • Therefore, f(f(2))=f(5)=1f(f(2)) = f(5) = 1.
  4. For g(g(1))g(g(1)):

    • First, find g(1)g(1) on the graph of g(x)g(x). When x=1x = 1, g(1)=3g(1) = 3.
    • Next, use g(3)g(3) on the graph of g(x)g(x). When x=3x = 3, g(3)=0g(3) = 0.
    • So, g(g(1))=g(3)=0g(g(1)) = g(3) = 0.

Final Answers:

  • f(g(5))=2f(g(5)) = 2
  • g(f(4))=1g(f(4)) = 1
  • f(f(2))=1f(f(2)) = 1
  • g(g(1))=0g(g(1)) = 0

Would you like further details on any of these steps?

Here are some related questions for additional practice:

  1. What is f(g(3))f(g(3))?
  2. What is g(f(1))g(f(1))?
  3. What is f(f(5))f(f(5))?
  4. What is g(g(4))g(g(4))?
  5. How would you find g(f(5))g(f(5))?

Tip: For composite functions like f(g(x))f(g(x)), evaluate the inner function first, then use its result to find the value of the outer function.

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Math Problem Analysis

Mathematical Concepts

Composite Functions
Function Evaluation
Graphs of Functions

Formulas

f(g(x)): Evaluate g(x) first, then f(result)
g(f(x)): Evaluate f(x) first, then g(result)

Theorems

Function Composition

Suitable Grade Level

Grades 9-12