Question

Give a rule for each piecewise-defined function. Also give the domain and range.

Accepted Answer

Step 1: Identify the different pieces of the piecewise function by looking at the graph and noting where the function changes its rule. Here, the graph has two distinct parts: one for x-values less than or equal to -1, and another for x-values greater than -1. Step 2: For the first piece (x ≤ -1), observe the graph's behavior. The graph is a horizontal line at y = -2 from x = -8 to x = -1, including the point (-1, -2) (solid dot). So, the rule for this piece is $$f(x) = -2$$ for $$x \leq -1. $$Step 3: For the second piece (x \u003e -1), the graph shows a curve starting just after x = -1 (open circle at (-1, 0)) and increasing through points like (1, 2) and (2, 3). This looks like a quadratic function. The curve resembles $$f(x) = x^2$ shifted down by 1 unit, so the rule is f(x$$) = $$x^2 - 1$ for x$$ \u003e -1$$. Step 4: Write the piecewise function combining both parts:

f(x$$) = \begin{cases} 
 -2 & 	ext{if } x \leq -1 \ 
 x^2 - 1 & 	ext{if } x \u003e -1 
 \end{cases}
$$ Step 5: Determine the domain and range. The domain is all x-values covered by the graph, which is from -8 to 2 (including -8 and 2). The range is the set of y-values the function takes: from -2 (lowest value on the horizontal line) up to 3 (highest point on the curve). So, domain: $$[-8, 2]$$, range: $$[-2, 3]$$.

Give a rule for each piecewise-defined function. Also give the domain and range.

Key Concepts

Piecewise-Defined Functions

Domain and Range

Interpreting Graphs with Open and Closed Points