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Part 1
Let f and g be defined by the table to the right.  Find the following.
StartRoot f left parenthesis negative 4 right parenthesis minus f left parenthesis negative 3 right parenthesis EndRoot minus left bracket g left parenthesis negative 1 right parenthesis right bracket squared plus f left parenthesis negative 5 right parenthesis divided by g left parenthesis negative 1 right parenthesis times g left parenthesis negative 4 right parenthesis
x
f left parenthesis x right parenthesis
g left parenthesis x right parenthesis
negative 5
2
0
negative 4
8
4
negative 3
 negative 1
1
negative 2
 negative 4
negative 8
negative 1
 0
negative 2
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Part 1
StartRoot f left parenthesis negative 4 right parenthesis minus f left parenthesis negative 3 right parenthesis EndRoot minus left bracket g left parenthesis negative 1 right parenthesis right bracket squared plus f left parenthesis negative 5 right parenthesis divided by g left parenthesis negative 1 right parenthesis times g left parenthesis negative 4 right parenthesisequals
  
enter your response here
​(Simplify your​ answer.)

Let's break the given expression down and use the values from the table for \( f(x) \) and \( g(x) \):

We need to find the following expression:

\[
\sqrt{f(-4) - f(-3)} - [g(-1)]^2 + \frac{f(-5)}{g(-1) \cdot g(-4)}
\]

### Step 1: Evaluate \( f(-4) - f(-3) \)
From the table:
- \( f(-4) = 8 \)
- \( f(-3) = -1 \)

So, \( f(-4) - f(-3) = 8 - (-1) = 8 + 1 = 9 \).

Now, take the square root:
\[
\sqrt{f(-4) - f(-3)} = \sqrt{9} = 3
\]

### Step 2: Evaluate \( [g(-1)]^2 \)
From the table:
- \( g(-1) = 0 \)

So:
\[
[g(-1)]^2 = 0^2 = 0
\]

### Step 3: Evaluate \( \frac{f(-5)}{g(-1) \cdot g(-4)} \)
From the table:
- \( f(-5) = 2 \)
- \( g(-1) = 0 \)
- \( g(-4) = 4 \)

So:
\[
g(-1) \cdot g(-4) = 0 \cdot 4 = 0
\]

However, dividing by 0 is undefined. Therefore, the entire expression is **undefined**.

### Final Answer:
The expression is undefined because we cannot divide by zero.

Let me know if you want further details or have any questions!

Here are 5 related questions:
1. What is the domain of a function?
2. How do you determine when a function is undefined?
3. What does it mean to take the square root of a negative number?
4. How do you simplify complex mathematical expressions involving square roots?
5. How can we use a table of values to evaluate functions?

**Tip**: Always check for undefined operations like division by zero to avoid incorrect conclusions.

Let f and g be defined by the table to the right. Find the following. StartRoot f left parenthesis negative 4 right parenthesis minus f left parenthesis negative 3 right parenthesis EndRoot minus left bracket g left parenthesis negative 1 right parenthesis right bracket squared plus f left parenthesis negative 5 right parenthesis divided by g left parenthesis negative 1 right parenthesis times g left parenthesis negative 4 right parenthesis

Explore a detailed solution to evaluating functions using values from a table, involving square roots, arithmetic operations, and recognizing undefined expressions. This problem is ideal for students in Grades 9-11 who are learning about function evaluation and division by zero.

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