Graphing Piecewise Functions

Problem: Graph and analyze the piecewise function:

\[ f(x) = \begin{cases} -x, & x \le 2 \\ x, & x > 2 \end{cases} \]

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Solution:

• Domain: All real numbers \( (-\infty, +\infty) \), since a formula is defined for every value of \( x \).
• Sketching: Graph \( y = -x \) for \( x \le 2 \) and \( y = x \) for \( x > 2 \).

Problem: Analyze the piecewise constant function:

\[ f(x) = \begin{cases} 2, & x > -3 \\ -5, & x < -3 \end{cases} \]

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Solution:

• Domain: All real numbers except \( x = -3 \).
• Range: \(\{-5, 2\}\).

Problem: Rewrite and graph \( f(x) = |x| \) as a piecewise function.

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Solution:

Using the definition of absolute value:

\[ f(x) = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases} \]

• Domain: All real numbers \( (-\infty, +\infty) \).
• Range: \( [0, +\infty) \).

Problem: Express \( f(x) = |x + 6| \) as a piecewise function and sketch.

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Solution:

\[ f(x) = \begin{cases} x + 6, & x \ge -6 \\ -(x + 6), & x < -6 \end{cases} \]

• Domain: All real numbers \( (-\infty, +\infty) \).

Problem: Analyze the function:

\[ f(x) = \begin{cases} x^2 - 3, & x \le -10 \\ -2x + 1, & -10 < x \le -2 \\ -x^3, & 2 < x < 4 \\ \ln x, & x > 4 \end{cases} \]

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Solution:

• Domain Note: The function is undefined on the interval \((-2, 2]\) and at \(x = 4\).

Problem: Find the domain and range, and sketch the graph of:

\[ f(x) = \begin{cases} -1, & x \le -2 \\ 2, & x > -2 \end{cases} \]

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Solution:

• Domain: All real numbers \( (-\infty, +\infty) \).
• Range: \(\{-1, 2\}\).

Problem: Find the domain, range, and sketch the graph of:

\[ f(x) = \begin{cases} x^2 + 1, & x < 2 \\ -x + 3, & x \ge 2 \end{cases} \]

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Solution:

• Domain: All real numbers \( (-\infty, +\infty) \).
• Range: \( (-\infty, +\infty) \).

Problem: Graph and analyze:

\[ f(x) = \begin{cases} \frac{1}{x}, & x < 0 \\ e^{-x}, & x \ge 0 \end{cases} \]

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Solution:

• Horizontal Asymptote: \( y = 0 \).

Problem: Find domain, range, and graph:

\[ f(x) = \begin{cases} -1, & x \le -1 \\ 1, & -1 < x \le 1 \\ x, & x > 1 \end{cases} \]

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Solution:

• Domain: All real numbers \( (-\infty, +\infty) \).
• Range: \(\{-1\} \cup [1, +\infty)\).