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sketch a graph of $f(x) = \\begin{cases} 2 & \\text{if } x \\leq -1 \\\…

The graph consists of: - A horizontal line segment at $y = 2$ for $x \leq -1$ (with a closed dot at $(-1, 2)$). - A line segment from $(-1, 2)$ (open dot, but the first segment ha…

Categoría: calculus Actualizado: 2026-02-04

Pregunta

sketch a graph of $f(x) = \\begin{cases} 2 & \\text{if } x \\leq -1 \\\\ -x + 1 & \\text{if } -1 < x \\leq 1 \\\\ 3 & \\text{if } x > 1 \\end{cases}$

notes: only use dots if two parts of the function dont connect.

question help: video 1 video 2
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Pasos de solución

  1. Understand the question

    sketch a graph of $f(x) = \\begin{cases} 2 & \\text{if } x \\leq -1 \\\\ -x + 1 & \\text{if } -1 < x \\leq 1 \\\\ 3 & \\text{if } x > 1 \\end{cases}$

    notes: only use dots if two parts of the function dont connect.

    question help: video 1 video 2
    calculator
    submit question

  2. Explanation

    Step1: Analyze the first piece ($x \leq -1$)

    The first part of the function is $f(x) = 2$ when $x \leq -1$. This is a horizontal line. We can plot a closed dot (since $x = -1$ is included) at $(-1, 2)$ and draw a horizontal line to the left (for all $x$ values less than or equal to -1).

    Step2: Analyze the second piece ($-1 < x \leq 1$)

    The second part is $f(x) = -x + 1$ for $-1 < x \leq 1$. This is a linear function. Let's find two points:

    • When $x = -1$ (but not included, so open dot), $f(-1) = -(-1) + 1 = 2$. Wait, but at $x = -1$, the first piece is 2 (closed dot), and the second piece at $x = -1$ would be 2 (open dot). So at $x = -1$, the closed dot from the first piece is there.
    • When $x = 1$ (included, closed dot), $f(1) = -1 + 1 = 0$. So we have a closed dot at $(1, 0)$ and an open dot at $(-1, 2)$ (but the first piece already has a closed dot there, so the second piece starts with an open dot at $(-1, 2)$ and goes to $(1, 0)$.

    Step3: Analyze the third piece ($x > 1$)

    The third part is $f(x) = 3$ when $x > 1$. This is a horizontal line. We plot an open dot at $(1, 3)$ (since $x = 1$ is not included here) and draw a horizontal line to the right (for all $x$ values greater than 1).

    Now, to sketch the graph:

    • For $x \leq -1$: Horizontal line at $y = 2$, closed dot at $(-1, 2)$.
    • For $-1 < x \leq 1$: Line from open dot $(-1, 2)$ (but the first piece has a closed dot there, so the second piece starts with an open dot, but visually, since the first piece is already at $(-1, 2)$, we can draw the line from $(-1, 2)$ (treating it as an open dot for the second piece, but the first piece's closed dot is there) to $(1, 0)$ (closed dot).
    • For $x > 1$: Horizontal line at $y = 3$, open dot at $(1, 3)$.
  3. Final answer

    The graph consists of:

    • A horizontal line segment at $y = 2$ for $x \leq -1$ (with a closed dot at $(-1, 2)$).
    • A line segment from $(-1, 2)$ (open dot, but the first segment has a closed dot there) to $(1, 0)$ (closed dot) for $-1 < x \leq 1$.
    • A horizontal line segment at $y = 3$ for $x > 1$ (with an open dot at $(1, 3)$).

    (Note: Since the question is to sketch the graph, the final answer is the graphical representation as described above. If you need to draw it, use the tools: for the first part, use the horizontal line tool from the left to $x = -1$ (closed dot at $(-1, 2)$); for the second part, use the line tool from $(-1, 2)$ (open dot) to $(1, 0)$ (closed dot); for the third part, use the horizontal line tool from $x = 1$ (open dot at $(1, 3)$) to the right.)

Respuesta

Explanation

Step1: Analyze the first piece ($x \leq -1$)

The first part of the function is $f(x) = 2$ when $x \leq -1$. This is a horizontal line. We can plot a closed dot (since $x = -1$ is included) at $(-1, 2)$ and draw a horizontal line to the left (for all $x$ values less than or equal to -1).

Step2: Analyze the second piece ($-1 < x \leq 1$)

The second part is $f(x) = -x + 1$ for $-1 < x \leq 1$. This is a linear function. Let's find two points:

  • When $x = -1$ (but not included, so open dot), $f(-1) = -(-1) + 1 = 2$. Wait, but at $x = -1$, the first piece is 2 (closed dot), and the second piece at $x = -1$ would be 2 (open dot). So at $x = -1$, the closed dot from the first piece is there.
  • When $x = 1$ (included, closed dot), $f(1) = -1 + 1 = 0$. So we have a closed dot at $(1, 0)$ and an open dot at $(-1, 2)$ (but the first piece already has a closed dot there, so the second piece starts with an open dot at $(-1, 2)$ and goes to $(1, 0)$.

Step3: Analyze the third piece ($x > 1$)

The third part is $f(x) = 3$ when $x > 1$. This is a horizontal line. We plot an open dot at $(1, 3)$ (since $x = 1$ is not included here) and draw a horizontal line to the right (for all $x$ values greater than 1).

Now, to sketch the graph:

  • For $x \leq -1$: Horizontal line at $y = 2$, closed dot at $(-1, 2)$.
  • For $-1 < x \leq 1$: Line from open dot $(-1, 2)$ (but the first piece has a closed dot there, so the second piece starts with an open dot, but visually, since the first piece is already at $(-1, 2)$, we can draw the line from $(-1, 2)$ (treating it as an open dot for the second piece, but the first piece's closed dot is there) to $(1, 0)$ (closed dot).
  • For $x > 1$: Horizontal line at $y = 3$, open dot at $(1, 3)$.

Answer

The graph consists of:

  • A horizontal line segment at $y = 2$ for $x \leq -1$ (with a closed dot at $(-1, 2)$).
  • A line segment from $(-1, 2)$ (open dot, but the first segment has a closed dot there) to $(1, 0)$ (closed dot) for $-1 < x \leq 1$.
  • A horizontal line segment at $y = 3$ for $x > 1$ (with an open dot at $(1, 3)$).

(Note: Since the question is to sketch the graph, the final answer is the graphical representation as described above. If you need to draw it, use the tools: for the first part, use the horizontal line tool from the left to $x = -1$ (closed dot at $(-1, 2)$); for the second part, use the line tool from $(-1, 2)$ (open dot) to $(1, 0)$ (closed dot); for the third part, use the horizontal line tool from $x = 1$ (open dot at $(1, 3)$) to the right.)

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Question Analysis

Subject mathematics
Sub Subject calculus
Education Level high school
Difficulty unspecified
Question Type with chart
Multi Question No
Question Count 1
Analysis Status completed
Analyzed At 2026-02-04T16:59:45

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sketch a graph of $f(x) = \\begin{cases} 2 & \\text{if } x \\leq -1 \\\\ -x + 1 & \\text{if } -1 < x \\leq 1 \\\\ 3 & \\text{if } x > 1 \\end{cases}$

notes: only use dots if two parts of the function dont connect.

question help: video 1 video 2
calculator
submit question

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