Graph each function. ƒ(x) = {|x| if x < 3 , 6-x if x ≥ 3}

Verified step by step guidance

1. Identify the piecewise function: $f(x)=|4x|$ for $x<4$ and $f(x)=-x+20$ for $x\ge 4$.
2. For $x<4$, graph $f(x)=|4x|$. This is a V-shaped graph with a vertex at the origin, but only consider the portion where $x<4$.
3. For $x\ge 4$, graph $f(x)=-x+20$. This is a straight line with a slope of -1 and a y-intercept of 20, starting from $x=4$.
4. Determine the point of intersection at $x=4$ by calculating $f(4)$ for both pieces: $f(4)=|4\times 4|=16$ and $f(4)=-4+20=16$.
5. Plot the graph by combining both pieces, ensuring continuity at $x=4$ where both pieces meet at the point (4, 16).

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two distinct rules: one for x values less than 4 and another for x values greater than or equal to 4. Understanding how to evaluate these functions at specific points is crucial for graphing them accurately.

Absolute Value

The absolute value function, denoted as |x|, represents the distance of x from zero on the number line, always yielding a non-negative result. In the context of the piecewise function, |4x| will affect the output for x values less than 4, creating a V-shaped graph that opens upwards, which is essential for understanding the behavior of the function in that interval.

Graphing Techniques

Graphing piecewise functions requires plotting each segment according to its defined interval. For f(x), one must graph |4x| for x < 4 and -x + 20 for x ≥ 4. Understanding how to transition between these segments and where to place open or closed dots on the graph is vital for accurately representing the function's behavior across its entire domain.