The problem in the image asks you to evaluate $(f \circ g)(x)$, which means you're finding the composition of the functions $f$ and $g$ at specific values of $x$.

Understanding the Problem

The composition $(f \circ g)(x)$ is defined as $f(g(x))$. To find this, follow these steps:

1. Look up the value of $g(x)$ from the given table.
2. Use the result from step 1 to find $f(g(x))$ by looking it up in the table for $f(x)$.

Given Tables

Here are the tables given in the problem:

x & 0 & 1 & 2 & 3 \\ \hline f(x) & 4 & 2 & 3 & 5 \\ g(x) & 1 & 2 & 3 & 0 \\ \end{array}$$ ### Steps to Solve 1. **Find $$g(x)$$:** - For $$x = 0$$, $$g(0) = 1$$ - For $$x = 1$$, $$g(1) = 2$$ - For $$x = 2$$, $$g(2) = 3$$ - For $$x = 3$$, $$g(3) = 0$$ 2. **Find $$f(g(x))$$:** - $$f(g(0)) = f(1) = 2$$ - $$f(g(1)) = f(2) = 3$$ - $$f(g(2)) = f(3) = 5$$ - $$f(g(3)) = f(0) = 4$$ ### Final Answer The values of $$(f \circ g)(x)$$ for each $$x$$ are: - $$(f \circ g)(0) = 2$$ - $$(f \circ g)(1) = 3$$ - $$(f \circ g)(2) = 5$$ - $$(f \circ g)(3) = 4$$ Would you like more details, or do you have any other questions? Here are five related questions: 1. How would you determine $$(g \circ f)(x)$$ for the same tables? 2. What is the definition of function composition? 3. Can the composition of two functions be commutative? 4. How can you determine if a function is one-to-one using a table? 5. What happens to $$f(g(x))$$ if $$g(x)$$ is a constant function? **Tip:** When dealing with function composition, always start from the innermost function and work your way outward!