Math / Geometry

Question6. Sketch the graph of the piecewise defined function. [7 points] $f(x)=\left\{\begin{array}{c} -x-4 \text { if } x \leq-3 \\ x-1 \text { if }-3<x \leq 3 \\ 3 \text { if } x>3 \end{array}\right.$

Studdy Solution

STEP 1

1. The function $f(x)$ is defined piecewise with three different expressions.
2. Each piece of the function is defined over a specific interval of $x$ .

STEP 2

1. Identify the intervals and corresponding expressions for the piecewise function.
2. Sketch each piece of the function over its respective interval.
3. Combine the pieces to form the complete graph.

STEP 3

Identify the intervals and corresponding expressions for the piecewise function:
- For $x \leq -3$ , $f(x) = -x - 4$ . - For $-3 < x \leq 3$ , $f(x) = x - 1$ . - For $x > 3$ , $f(x) = 3$ .

STEP 4

Sketch each piece of the function over its respective interval:
- For $x \leq -3$ , sketch the line $f(x) = -x - 4$ . This is a line with a slope of $-1$ and a y-intercept of $-4$ . It continues to the left indefinitely. - For $-3 < x \leq 3$ , sketch the line $f(x) = x - 1$ . This is a line with a slope of $1$ and a y-intercept of $-1$ . It starts just after $x = -3$ and ends at $x = 3$ .
- For $x > 3$ , sketch the horizontal line $f(x) = 3$ . This line is constant and extends to the right indefinitely starting just after $x = 3$ .

STEP 5

Combine the pieces to form the complete graph:
- Ensure the transition between pieces is clear. At $x = -3$ , the function transitions from $f(x) = -x - 4$ to $f(x) = x - 1$ . Check the values at $x = -3$ to ensure continuity or note any discontinuity. - At $x = 3$ , the function transitions from $f(x) = x - 1$ to $f(x) = 3$ . Check the values at $x = 3$ to ensure continuity or note any discontinuity.
The graph should now be sketched with the appropriate lines and transitions between the intervals.

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