Use this interactive piecewise functions calculator to quickly evaluate $f(x)$ for any input value $x$ based on the predefined conditional function structure.
Piecewise Functions Calculator
Function Definition:
$$ f(x) = \begin{cases} x^2 – 4 & \text{if } x < -2 \\ 3x + 2 & \text{if } -2 \le x \le 5 \\ 10 - x & \text{if } x > 5 \end{cases} $$Calculated Result, f(x):
Piecewise Functions Calculator Formula
The general structure of a piecewise function $f(x)$ is:
$$ f(x) = \begin{cases} \text{Formula}_1 & \text{if } \text{Condition}_1 \\ \text{Formula}_2 & \text{if } \text{Condition}_2 \\ \dots & \dots \end{cases} $$Variables
- Variable Value (x): The independent variable for which the function is to be evaluated. This determines which piece of the function’s definition is used.
- Formula: The algebraic expression applied to $x$ when the associated condition is met.
- Condition: The interval or specific value of $x$ that dictates which formula is active.
Related Calculators
Derivative Calculator Integral Area Calculator Limit Evaluator Polynomial Root FinderWhat is a Piecewise Function?
A piecewise function is a function defined by multiple sub-functions, where each sub-function applies to a different interval or condition of the independent variable $x$. These functions are essential in modeling real-world scenarios where different mathematical rules apply based on specific conditions, such as tax brackets, shipping costs, or velocity over time.
The crucial aspect of a piecewise function is evaluating the input value $x$ against the set of conditions to determine which formula is correct. Only one condition can be true for any given $x$. If the conditions are defined correctly, there should be no ambiguity or overlap, though special attention is often paid to the boundary points where the function transitions from one formula to another.
How to Calculate Piecewise Functions (Example)
Using the defined function $f(x)$ above, let’s calculate $f(3)$:
- Check Condition 1: Is $3 < -2$? (False)
- Check Condition 2: Is $-2 \le 3 \le 5$? (True)
- Apply Formula 2: Since the second condition is true, we use $f(x) = 3x + 2$.
- Substitute: $f(3) = 3(3) + 2 = 9 + 2 = 11$.
- Result: $f(3) = 11$.
Frequently Asked Questions (FAQ)
The domain of a piecewise function is the union of the intervals of its conditions. For the function to be valid, these conditions must cover all the desired input values, often the entire set of real numbers ($\mathbb{R}$).
Yes, a piecewise function can be continuous if all its individual sub-functions are continuous over their respective domains, AND the function’s value is the same at every boundary point where the conditions change.
If conditions overlap (e.g., $x \le 5$ and $x \ge 3$), the function is technically ill-defined unless both formulas give the exact same result in the overlapping region. Well-defined piecewise functions ensure conditions are mutually exclusive.
Yes, the calculator is programmed to follow the precise logical flow of the function’s conditions and mathematical formulas, guaranteeing an accurate evaluation for any valid numerical input $x$.