Math Problem Statement

Use the table of values to evaluate the following expressions: f(g(0)), g(f(1)), f(f(9)), and g(g(8)).

Solution

Let's evaluate the given expressions using the table of values provided.

1. f(g(0))f(g(0))

From the table:

  • g(0)=5g(0) = 5 Now, use this value to find f(5)f(5):
  • f(5)=1f(5) = 1

So, f(g(0))=1f(g(0)) = 1.

2. g(f(1))g(f(1))

From the table:

  • f(1)=0f(1) = 0 Now, use this value to find g(0)g(0):
  • g(0)=5g(0) = 5

So, g(f(1))=5g(f(1)) = 5.

3. f(f(9))f(f(9))

From the table:

  • f(9)=6f(9) = 6 Now, use this value to find f(6)f(6):
  • f(6)=9f(6) = 9

So, f(f(9))=9f(f(9)) = 9.

4. g(g(8))g(g(8))

From the table:

  • g(8)=3g(8) = 3 Now, use this value to find g(3)g(3):
  • g(3)=6g(3) = 6

So, g(g(8))=6g(g(8)) = 6.

Final Results:

  • f(g(0))=1f(g(0)) = 1
  • g(f(1))=5g(f(1)) = 5
  • f(f(9))=9f(f(9)) = 9
  • g(g(8))=6g(g(8)) = 6

Would you like more details or further explanations on any of these steps?

Here are some related questions to think about:

  1. What would g(g(0))g(g(0)) be based on the table?
  2. How do nested functions like f(f(x))f(f(x)) or g(g(x))g(g(x)) work conceptually?
  3. Can you identify an input xx where f(x)=g(x)f(x) = g(x)?
  4. What would the result be for f(g(9))f(g(9))?
  5. How can you describe the difference between f(x)f(x) and g(x)g(x) based on the values?

Tip: Always make sure to evaluate inner functions first when dealing with nested functions!

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Math Problem Analysis

Mathematical Concepts

Functions
Composition of Functions

Formulas

f(g(x)) = f evaluated at g(x)
g(f(x)) = g evaluated at f(x)

Theorems

Composition of Functions Theorem

Suitable Grade Level

Grades 9-11