Math Problem Statement

Given a table with functions f(x) and g(x) defined for certain values of x, find the values of: (a) g(f(1)), (b) f(g(4)), (c) f(f(3)).

Solution

The image contains a table with functions f(x)f(x) and g(x)g(x) defined for certain values of xx. The problem asks to find the values for the following:

  • (a) g(f(1))g(f(1))
  • (b) f(g(4))f(g(4))
  • (c) f(f(3))f(f(3))

Step-by-Step Solution:

Given Table:

x & f(x) & x & g(x) \\ \hline 1 & 4 & 1 & 2 \\ 2 & 3 & 2 & 3 \\ 3 & 1 & 3 & 4 \\ 4 & 2 & 4 & 5 \\ \end{array}$$ ### (a) $$g(f(1))$$ 1. First, find $$f(1)$$ from the table. $$f(1) = 4$$ 2. Now, find $$g(4)$$ using the result from the previous step. $$g(4) = 5$$ Thus, $$g(f(1)) = 5$$. ### (b) $$f(g(4))$$ 1. First, find $$g(4)$$ from the table. $$g(4) = 5$$ 2. Now, find $$f(5)$$. However, the table does not provide a value for $$f(5)$$, meaning $$f(5)$$ is not defined based on the given data. Thus, $$f(g(4))$$ is **undefined**. ### (c) $$f(f(3))$$ 1. First, find $$f(3)$$ from the table. $$f(3) = 1$$ 2. Now, find $$f(1)$$ using the result from the previous step. $$f(1) = 4$$ Thus, $$f(f(3)) = 4$$. ### Summary: - (a) $$g(f(1)) = 5$$ - (b) $$f(g(4))$$ is **undefined**. - (c) $$f(f(3)) = 4$$ Would you like further details on this, or do you have any other questions? ### Related Questions: 1. What is the value of $$g(g(2))$$? 2. What would $$f(f(1))$$ be? 3. How do you interpret $$g(g(x))$$ for a given $$x$$? 4. What if the table included $$f(5) = 6$$? How would that change part (b)? 5. Can a function be defined such that both $$f(x)$$ and $$g(x)$$ are inverses of each other? ### Tip: When dealing with nested functions like $$f(f(x))$$ or $$g(f(x))$$, always solve from the innermost function outward.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Function Evaluation
Algebra

Formulas

f(x) and g(x) values

Theorems

-

Suitable Grade Level

Grades 9-12

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