Certainly! Let's work through the solution step-by-step for the function \( f(x) = -3|x-2|+6 \).
1. Understanding Absolute Value Function:
The absolute value function \( |x-2| \) produces a non-negative value. Specifically,
[tex]\[
|x-2| = \begin{cases}
x-2 & \text{if } x \geq 2 \\
-(x-2) & \text{if } x < 2
\end{cases}
\][/tex]
2. Piecewise Definition of the Function:
Now we need to define the function \( f(x) = -3|x-2| + 6 \) in a piecewise manner based on the absolute value.
[tex]\[
f(x) = \begin{cases}
-3(x-2) + 6 & \text{if } x \geq 2 \\
-3(-(x-2)) + 6 & \text{if } x < 2
\end{cases}
\][/tex]
3. Simplifying Each Piece:
- For \( x \geq 2 \):
[tex]\[
f(x) = -3(x-2) + 6 = -3x + 6 + 6 = -3x + 12
\][/tex]
- For \( x < 2 \):
[tex]\[
f(x) = -3(-(x-2)) + 6 = -3(-x + 2) + 6 = 3x - 6 + 6 = 3x
\][/tex]
4. Final Piecewise Function:
Assemble the simplified expressions into a single piecewise function:
[tex]\[
f(x) = \begin{cases}
-3x + 12 & \text{if } x \geq 2 \\
3x & \text{if } x < 2
\end{cases}
\][/tex]
5. Graphical Interpretation:
The graph of \( f(x) \) consists of two linear segments:
- A line with a slope of -3 starting from the point \( (2, 6) \) and extending to the right (for \( x \geq 2 \)).
- A line with a slope of 3 starting from \( (2, 6) \) and extending to the left (for \( x < 2 \)).
6. Critical Point:
The transition point at \( x = 2 \) is where the absolute value function changes its behavior. At this point, \( f(x) \) is continuous and smooth given that both pieces meet at \( (2, 6) \).
In summary, the function [tex]\( f(x) = -3|x-2| + 6 \)[/tex] can be represented and analyzed using a piecewise linear approach as detailed above. This yields the final representation and provides insights into its behavior for different values of [tex]\( x \)[/tex].
