Piecewise Functions, Increasing/Decreasing/Constant Functions, Domain & Range, and Relative Extrema

Piecewise Functions

Definition and Structure

Piecewise functions are functions defined by different expressions for different intervals of the domain. They are useful for modeling situations where a rule changes depending on the input value.

• Piecewise Function: A function composed of multiple sub-functions, each defined on a specific interval.

• Notation: Typically written as: $f(x) = \begin{cases} \text{expression 1, for } x \text{ in interval 1} \\ \text{expression 2, for } x \text{ in interval 2} \end{cases}$

• Graphing: Plot each piece on its respective interval, paying attention to open or closed endpoints.

Example

• Suppose $g(x)$ is defined as: $g(x) = \begin{cases} 5x + 3, & x < 3 \\ x^3, & x \geq 3 \end{cases}$

• For $x = 2$, $g(2) = 5 \times 2 + 3 = 13$

• For $x = 3$, $g(3) = 3^3 = 27$

Additional info: When graphing, use open circles for endpoints not included, and closed circles for included endpoints.

Increasing, Decreasing, and Constant Functions

Definitions and Identification

Understanding how a function behaves as its input changes is fundamental in precalculus. The function may increase, decrease, or remain constant over different intervals.

• Increasing: A function is increasing on an interval if, as $x$ moves from left to right, $f(x)$ increases.

• Decreasing: A function is decreasing on an interval if, as $x$ moves from left to right, $f(x)$ decreases.

• Constant: A function is constant on an interval if $f(x)$ remains the same for all $x$ in that interval.

• Interval Notation: Use parentheses for open intervals and brackets for closed intervals. Always use $x$-values to describe intervals.

Example

• Given a function $f(x)$, suppose:

  • Increasing on $(3, 5)$

  • Decreasing on $(-5, -1)$ and $(5, 0)$

  • Constant on $(-1, 3)$

Additional info: To determine these intervals, examine the graph from left to right and observe the direction of the curve.

Domain and Range

Definitions and Determination

The domain and range of a function describe the set of possible input and output values, respectively. These are fundamental concepts for understanding the scope of a function.

• Domain: The set of all possible $x$-values (inputs) for which the function is defined.

• Range: The set of all possible $y$-values (outputs) the function can produce.

• Interval Notation: Used to express domain and range. For example, $[-3, 5]$ means all values from $-3$ to $5$, inclusive.

• Graphical Determination: Domain is read from left to right along the $x$-axis; range is read from bottom to top along the $y$-axis.

Example

• If a graph extends from $x = -3$ to $x = 5$$, then:$ If the graph's lowest $y$-value is $-3$ and highest is $0$, then: $\text{Range}: [-3, 0)$

Additional info: Use open intervals for values not included and closed intervals for included endpoints.

Relative Maxima and Minima

Definitions and Identification

Relative maxima and minima are the highest and lowest points (peaks and valleys) within a specific interval of a function. These are also known as turning points.

• Relative Maximum: A point where the function changes from increasing to decreasing; the $y$-value is higher than nearby points.

• Relative Minimum: A point where the function changes from decreasing to increasing; the $y$-value is lower than nearby points.

• Identification: Look for peaks (maxima) and valleys (minima) on the graph.

Example

• If $f(x)$ has a peak at $x = 2$ with $f(2) = 3$, then there is a relative maximum of $3$at$x = 2$.

• If $f(x)$ has a valley at $x = -3$ with $f(-3) = -1$, then there is a relative minimum of $-1$ at $x = -3$.

Additional info: Relative extrema are important for analyzing the behavior of functions and solving optimization problems.