What is Piecewise Functions?

Defining and evaluating functions with different rules on different intervals

How can I study Piecewise Functions effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

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What course covers Piecewise Functions?

Piecewise Functions is part of the AP Precalculus course on Study Mondo, specifically in the Function Fundamentals section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Piecewise Functions?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Piecewise Functions

What is a Piecewise Function?

A piecewise function is defined by different formulas on different parts of its domain.

General Form

\text{formula 1} & \text{if condition 1} \\ \text{formula 2} & \text{if condition 2} \\ \vdots & \vdots \end{cases}$$ ### Example $$f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \geq 0 \end{cases}$$ This means: - Use $f(x) = x^2$ when $x$ is negative - Use $f(x) = 2x + 1$ when $x$ is zero or positive ## Evaluating Piecewise Functions To evaluate $f(a)$ for a piecewise function: 1. Determine which condition $a$ satisfies 2. Use the formula for that piece 3. Calculate the value ### Example For the function above: - $f(-2) = (-2)^2 = 4$ (use first piece since $-2 < 0$) - $f(0) = 2(0) + 1 = 1$ (use second piece since $0 \geq 0$) - $f(3) = 2(3) + 1 = 7$ (use second piece since $3 \geq 0$) ## Continuity of Piecewise Functions A piecewise function is **continuous** at $x = a$ if: 1. $f(a)$ is defined 2. $\lim_{x \to a} f(x)$ exists 3. $\lim_{x \to a} f(x) = f(a)$ **Practical check:** The function is continuous if the pieces "connect" (no jump or gap). ### Checking Continuity at a Boundary For continuity at $x = c$ (a boundary point): - Evaluate the left piece at $x = c$: $\lim_{x \to c^-} f(x)$ - Evaluate the right piece at $x = c$: $\lim_{x \to c^+} f(x)$ - Check which piece includes $c$ to find $f(c)$ - If left limit = right limit = $f(c)$, the function is continuous ## Absolute Value as a Piecewise Function The absolute value function is a common piecewise function: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ ## Domain and Range - **Domain**: Usually all real numbers, unless specified otherwise - **Range**: Combine the ranges from all pieces (may need to sketch or analyze) ## Graphing Piecewise Functions 1. Graph each piece on its specified interval 2. Use open circles ($\circ$) for excluded endpoints 3. Use closed circles ($\bullet$) for included endpoints 4. Check for continuity at boundary points

Evaluate the piecewise function $f(x) = \begin{cases} x^2 + 1 & \text{if } x \leq -1 \\ 3x & \text{if } -1 < x < 2 \\ 5 & \text{if } x \geq 2 \end{cases}$ at , , and .

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

Evaluate at each point:

For $x = -1$ : Check which condition: $-1 \leq -1$ ✓ (first piece) $f(-1) = (-1)^2 + 1 = 1 + 1 = 2$

For $x = 0$ : Check which condition: $-1 < 0 < 2$ ✓ (second piece) $f(0) = 3(0) = 0$

For $x = 3$ : Check which condition: $3 \geq 2$ ✓ (third piece) $f(3) = 5$

Answers:

$f(-1) = 2$
$f(0) = 0$
$f(3) = 5$

Determine whether the piecewise function $g(x) = \begin{cases} x + 3 & \text{if } x < 1 \\ x^2 + 1 & \text{if } x \geq 1 \end{cases}$ is continuous at $x = 1$ .

Check continuity at $x = 1$ :

Step 1: Find $g(1)$ (which piece includes $x = 1$ ?) Since $x \geq$ includes 1, use second piece:

Find the value of $a$ that makes $h(x) = \begin{cases} 2x + a & \text{if } x < 3 \\ x^2 - 4 & \text{if } x \geq 3 \end{cases}$ continuous at $x = 3$ .

For continuity at $x = 3$ , we need: $\lim_{x \to 3^-} h(x) = \lim_{x \to 3^+} h(x) = h(3)$

A function is defined as f(x) = { x² + 1, if x < 0; 2x + 1, if x ≥ 0 }. Evaluate f(-2), f(0), and f(3).

Step 1: Evaluate f(-2): Is -2 < 0? Yes Use f(x) = x² + 1 f(-2) = (-2)² + 1 = 4 + 1 = 5

Step 2: Evaluate f(0): Is 0 < 0? No Is 0 ≥ 0? Yes Use f(x) = 2x + 1 f(0) = 2(0) + 1 = 1

Step 3: Evaluate f(3): Is 3 < 0? No Is 3 ≥ 0? Yes Use f(x) = 2x + 1 f(3) = 2(3) + 1 = 7

Answer: f(-2) = 5, f(0) = 1, f(3) = 7

For f(x) = { -x + 2, if x ≤ 1; x² - 2, if x > 1 }, determine if f is continuous at x = 1.

Step 1: Find f(1) using the appropriate piece: Since 1 ≤ 1, use f(x) = -x + 2 f(1) = -1 + 2 = 1

Step 2: Find the left-hand limit as x approaches 1: lim(x→1⁻) f(x) = lim(x→1⁻) (-x + 2) = -1 + 2 = 1

Step 3: Find the right-hand limit as x approaches 1: lim(x→1⁺) f(x) = lim(x→1⁺) (x² - 2) = 1² - 2 = -1

Step 4: Check continuity conditions: For continuity at x = 1: • lim(x→1⁻) f(x) = 1 • lim(x→1⁺) f(x) = -1 • f(1) = 1

Step 5: Conclusion: Since lim(x→1⁻) f(x) ≠ lim(x→1⁺) f(x), the function is NOT continuous at x = 1. There is a jump discontinuity at x = 1.

Answer: Not continuous (jump discontinuity at x = 1)

Piecewise Functions

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Piecewise Functions

What is a Piecewise Function?

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