First, sketch a graph of the following piecewise function.𝑓(𝑥)=⎧⎩⎨⎪⎪𝑥0𝑥𝑥<00≤𝑥<11≤𝑥

Break Down the Problem

1. Analyze the piecewise function $f(x)$:
   • $f(x) = x$ for $x < 0$
   • $f(x) = 0$ for $0 \leq x < 1$
   • $f(x) = 1$ for $x \geq 1$

Relevant Concepts

2. Understand how each piece of the function behaves in its respective domain:
   • For $x < 0$: The function is a linear function with a slope of 1, passing through the origin.
   • For $0 \leq x < 1$: The function is constant at $f(x) = 0$.
   • For $x \geq 1$: The function is constant at $f(x) = 1$.

Analysis and Detail

3. Determine key points to plot:
   • At $x < 0$: The line $f(x) = x$ approaches from the left.
   • At $x = 0$: $f(0) = 0$.
   • From $0$ to $1$: The function remains at $0$.
   • At $x = 1$: $f(1) = 1$.
   • For $x > 1$: The function remains at $1$.

Verify and Summarize

4. Summary of key points for the graph:
   • $(-\infty, 0)$: Line with slope 1 (linear).
   • $(0, 1)$: Horizontal line at $y = 0$.
   • $x \geq 1$: Horizontal line at $y = 1$.

Final Answer

To sketch the graph:

• Plot the line for $x < 0$ extending through the origin.
• Highlight that at $x = 0$ the value is 0 (including this point).
• Draw a horizontal line from $(0, 0)$ to $(1, 0)$.
• At $x = 1$, include the point $(1, 1)$, and continue the line horizontally to the right for $x > 1$ at $y = 1$.

The graph will display a transition from the line to the horizontal segments as described.