Evaluate the function at the indicated values. (If an answer is undefined, enter UNDEFINED.)
f(x+1) = 6|x − 4|

To evaluate the function \( f(x+1) = 6|x - 4| \) at given values, let’s break down the process:

### Step 1: Understanding the function
The function is \( f(x+1) = 6|x - 4| \), where \( |x - 4| \) represents the absolute value of \( x - 4 \). 

### Step 2: Evaluating the function at different values
Let’s substitute different values for \( x+1 \) (the argument of the function) and compute the corresponding results.

#### 1. **For \( f(0) \):**
We set \( x+1 = 0 \). Therefore, \( x = -1 \).
Now substitute \( x = -1 \) into the function:
\[
f(0) = 6|-1 - 4| = 6| -5 | = 6 \times 5 = 30
\]
So, \( f(0) = 30 \).

#### 2. **For \( f(2) \):**
We set \( x+1 = 2 \). Therefore, \( x = 1 \).
Now substitute \( x = 1 \) into the function:
\[
f(2) = 6|1 - 4| = 6| -3 | = 6 \times 3 = 18
\]
So, \( f(2) = 18 \).

#### 3. **For \( f(5) \):**
We set \( x+1 = 5 \). Therefore, \( x = 4 \).
Now substitute \( x = 4 \) into the function:
\[
f(5) = 6|4 - 4| = 6|0| = 6 \times 0 = 0
\]
So, \( f(5) = 0 \).

#### 4. **For \( f(6) \):**
We set \( x+1 = 6 \). Therefore, \( x = 5 \).
Now substitute \( x = 5 \) into the function:
\[
f(6) = 6|5 - 4| = 6|1| = 6 \times 1 = 6
\]
So, \( f(6) = 6 \).

#### 5. **For \( f(8) \):**
We set \( x+1 = 8 \). Therefore, \( x = 7 \).
Now substitute \( x = 7 \) into the function:
\[
f(8) = 6|7 - 4| = 6|3| = 6 \times 3 = 18
\]
So, \( f(8) = 18 \).

### Final Answers:
- \( f(0) = 30 \)
- \( f(2) = 18 \)
- \( f(5) = 0 \)
- \( f(6) = 6 \)
- \( f(8) = 18 \)

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions to deepen your understanding:
1. What is the general behavior of absolute value functions?
2. How does shifting the input of a function by 1 affect its graph?
3. What happens to the function if we change the constant multiplier (e.g., from 6 to 3)?
4. How would you solve for \( x \) if the function was set equal to a specific value, say \( f(x+1) = 12 \)?
5. How can you find the vertex of the absolute value function in this case?

**Tip**: Absolute value functions create a "V" shape on a graph, with a turning point at the value that makes the expression inside the absolute value zero.

Evaluate the function at the indicated values. (If an answer is undefined, enter UNDEFINED.) f(x+1) = 6|x − 4|

Learn how to evaluate the absolute value function f(x+1) = 6|x − 4| at various values of x. This problem explains the key concepts of absolute value and function evaluation, suitable for students in Grades 9-10.

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