https://file.aiquickdraw.com/mathbot/file/17264583656067f8ti8az.jpg

Let's work through the problem one part at a time using the functions and tables provided in the image.

We have the following information:

1. **Function \( f(x) \)** given as a set of pairs:
   - \( f(x) = \{(0,1), (1,0), (2,5), (3,2), (4,3), (5,4)\} \)
   This is a lookup table, where for example, \( f(0) = 1 \), \( f(1) = 0 \), etc.

2. **Function \( g(x) \)** represented as:
   - \( g(x) \) is given as a table with \( x \) and corresponding \( y \) values:
   \[
   \begin{array}{|c|c|}
   x & y \\
   \hline
   -3 & -6 \\
   -1 & 6 \\
   0 & 3 \\
   3 & 5 \\
   4 & 8 \\
   5 & 11 \\
   6 & 3 \\
   \end{array}
   \]
   For example, \( g(3) = 5 \), \( g(5) = 11 \), etc.

3. **Function \( h(x) \)** defined as:
   - \( h(x) = \sqrt{x - 2} \)
   This is a square root function, so to compute \( h(x) \), we use this formula.

4. **Function \( j(x) \)** is graphed. The graph provides a way to evaluate \( j(x) \) by checking the corresponding \( y \)-value for a given \( x \)-value.

### Let's now answer the questions one by one:

---

**A. \( h(g(5)) \)**

First, we compute \( g(5) \) using the table for \( g(x) \).  
From the table, \( g(5) = 11 \).

Next, compute \( h(11) \) using the formula \( h(x) = \sqrt{x - 2} \):
\[
h(11) = \sqrt{11 - 2} = \sqrt{9} = 3
\]

So, \( h(g(5)) = 3 \).

---

**B. \( f(j(1)) \)**

To solve this, we need to first find \( j(1) \) from the graph of \( j(x) \). Looking at the graph, when \( x = 1 \), \( j(x) = 2 \).

Next, we compute \( f(2) \) using the set of pairs for \( f(x) \). From the set, \( f(2) = 5 \).

So, \( f(j(1)) = 5 \).

---

**C. \( j(h(6)) \)**

First, compute \( h(6) \) using the formula \( h(x) = \sqrt{x - 2} \):
\[
h(6) = \sqrt{6 - 2} = \sqrt{4} = 2
\]

Now, use the graph to find \( j(2) \). From the graph, when \( x = 2 \), \( j(x) = 4 \).

So, \( j(h(6)) = 4 \).

---

**D. \( g(f(3)) \)**

First, compute \( f(3) \) using the set of pairs for \( f(x) \). From the set, \( f(3) = 2 \).

Next, compute \( g(2) \) using the table for \( g(x) \). Unfortunately, the table does not directly give \( g(2) \). Based on this, we might infer that \( g(2) \) isn't defined here, or we need more information.

Thus, **\( g(f(3)) \) cannot be determined** with the given information from the table.

---

Would you like further details or clarification? Here are five related questions to deepen understanding:

1. What happens if \( h(x) \) is undefined for certain inputs?
2. How does the graph of \( j(x) \) help with interpreting the function's behavior?
3. Can you find \( g(x) \) values through interpolation if they're missing from the table?
4. How does the square root function behave when \( x < 2 \)?
5. What is the general approach to solving compositions of functions like these?

**Tip**: Always check if the input to a function lies within its domain to ensure you're not dealing with undefined values.

Use the group of functions below to answer the following questions: A. h(g(5)), B. f(j(1)), C. j(h(6)), D. g(f(3))

Learn how to solve function compositions using provided tables, graphs, and formulas. This example covers key mathematical concepts like composition of functions, square root functions, and graph interpretation. Ideal for high school students (Grades 10-12).

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