IntroductionFinding the domain of the following piecewise function is a common task in algebra and pre‑calculus, yet many students feel uncertain when a function is defined by multiple pieces. The domain refers to all real numbers x for which the function yields a valid output. Because a piecewise function changes its rule at certain breakpoints, the process requires careful inspection of each segment, verification of any restrictions (such as division by zero or square roots of negative numbers), and finally the combination of all permissible x values. This article will guide you step‑by‑step, using clear subheadings, bold highlights, and bullet‑point lists to make the procedure easy to follow and to ensure you can find the domain of the following piecewise function confidently.
Understanding Piecewise Functions
A piecewise function is defined by different expressions over disjoint intervals of the independent variable. The general form looks like:
[ f(x)= \begin{cases} \text{expression}_1 & \text{for } x \in I_1\[4pt] \text{expression}_2 & \text{for } x \in I_2\[4pt] ;;\vdots\[4pt] \text{expression}_n & \text{for } x \in I_n \end{cases} ]
Each interval (I_k) may be open, closed, half‑open, or bounded. The domain is the union of all x values that belong to any interval and satisfy any algebraic restrictions in the corresponding expression.
Key points to remember
- Interval notation (e.g., ((-\infty, -2]), ([0, 3))) tells you which endpoints are included.
- Restrictions such as denominators cannot be zero, radicands must be non‑negative, and logarithms require positive arguments.
- The final domain is the set of all x that satisfy both the interval condition and any algebraic constraints.
Steps to Determine the Domain
Below is a concise, numbered checklist you can follow whenever you need to find the domain of the following piecewise function.
- List all intervals defined for the piecewise function.
- Identify restrictions for each expression (denominator ≠ 0, radicand ≥ 0, argument > 0, etc.).
- Solve each restriction to obtain the set of x values that keep the expression valid.
- Intersect the solution sets from step 3 with the interval (I_k) for that piece.
- Take the union of the valid sets from all pieces.
- Write the final domain in interval notation (or set‑builder form) and double‑check for any missed endpoints.
Example Walkthrough
Consider the piecewise function:
[ f(x)= \begin{cases} \displaystyle \frac{1}{x-2} & \text{if } x < 0\[6pt] \sqrt{x+4} & \text{if } 0 \le x \le 5\[6pt] \ln(10-x) & \text{if } x > 5 \end{cases} ]
Step‑by‑step solution
- Intervals:
- Piece 1: (x < 0)
