Math Problem Statement

Evaluate g(f(6)) using the provided graphs for f(x) and g(x).

Solution

Let's solve the problem using the provided graphs for f(x)f(x) and g(x)g(x), and the given composition g(f(6))g(f(6)).

Step 1: Evaluate f(6)f(6)

From the graph of y=f(x)y = f(x) (left-hand side):

  • Locate x=6x = 6.
  • From the graph, the corresponding value of f(6)f(6) is 7.

Thus, f(6)=7f(6) = 7.


Step 2: Evaluate g(7)g(7)

Now, using the graph of y=g(x)y = g(x) (right-hand side):

  • Locate x=7x = 7.
  • From the graph, the corresponding value of g(7)g(7) is 0.

Thus, g(7)=0g(7) = 0.


Final Answer:

g(f(6))=g(7)=0g(f(6)) = g(7) = 0

Would you like more details or clarification? Let me know!

Related Questions:

  1. How would the solution change if f(6)f(6) were a different value, say 88?
  2. How is the process of finding g(f(6))g(f(6)) related to function composition in general?
  3. Can g(f(x))g(f(x)) ever equal 11 given these graphs? Why or why not?
  4. What is the domain of g(f(x))g(f(x)) in this scenario?
  5. What is the range of g(f(x))g(f(x)) when x[10,10]x \in [-10, 10]?

Tip:

When evaluating composite functions graphically, carefully match the xx-value on one graph to the yy-value on the other!

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Math Problem Analysis

Mathematical Concepts

Function Composition
Graphical Interpretation of Functions

Formulas

f(a) = b (from graph of f(x))
g(b) = c (from graph of g(x))

Theorems

Definition of Function Composition

Suitable Grade Level

Grades 9-12