Explain how to graph the given piecewise-defined function. Be sure to specify the type of endpoint each piece of the function wi

In certain cases, a function necessitates multiple formulas to achieve the desired outcome. An example is the absolute value function \displaystyle f\left(x\right)=|x|f(x)=∣x∣. This function applies to all real numbers and yields results that are non-negative, defining absolute value as the magnitude or modulus of a real number regardless of its sign. It indicates the distance from zero on the number line, requiring all outputs to be zero or greater.

\displaystyle f\left(x\right)=x\text{ if }x\ge 0f(x)=x if x≥0

\displaystyle f\left(x\right)=-x\text{ if }x<0f(x)=−x if x<0

Due to the need for two distinct operations, the absolute value function qualifies as a piecewise function: a function defined by several formulas for different sections of its domain.

Piecewise functions help describe scenarios where rules or relationships alter as the input crosses specific "boundaries." Business contexts often demonstrate this, such as when the cost per unit of an item decreases past a certain order quantity. The concept of tax brackets also illustrates piecewise functions. For instance, in a basic tax system where earnings up to $10,000 face a 10% tax, additional income incurs a 20% tax rate. Thus, the total tax on an income S would be 0.1S when \displaystyle {S}\leS≤ $10,000 and 1000 + 0.2 (S – $10,000) when S > $10,000.

Sample Response: To graph the equation f(x) = –x + 3 for x values less than 2, place an open circle at (2, 1) because the first piece’s domain excludes 2. Next, graph a horizontal line at f(x) = 3 for inputs from 2 to 4, marking a closed circle at (2, 3) and an open circle at (4, 3). Finally, sketch f(x) = 4 – 2x for x values starting at 4, ensuring a closed circle at (4, –4) since 4 belongs to the domain of the third segment.