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.