How can you find the domain and range of a piecewise function

How can you find the domain and range of a piecewise function? Use Desmos to give an example

The Correct Answer and Explanation is:

To find the domain and range of a piecewise function, you need to analyze each part of the function individually. A piecewise function is defined by different expressions over different intervals, so the domain and range need to be determined for each piece.

Steps to Find the Domain and Range:

1. Determine the Domain:
   • The domain of a function refers to the set of possible input values (x-values). For a piecewise function, you look at each individual piece and identify the x-values that it covers.
   • The domain is the union of all the intervals where the function is defined.
   • If a piece is only valid for a specific range of x-values, make sure you include those intervals when determining the full domain.
2. Determine the Range:
   • The range refers to the set of possible output values (y-values). For each piece of the piecewise function, calculate the corresponding y-values (based on the formula of that segment) and combine them.
   • Sometimes, you need to evaluate the function at the endpoints of the intervals to determine the possible y-values.

Example Using Desmos:

Let’s consider the following piecewise function:f(x)={x+2if x≤1−2x+5if 1<x≤4×2−3if x>4f(x) = \begin{cases} x + 2 & \text{if } x \leq 1 \\ -2x + 5 & \text{if } 1 < x \leq 4 \\ x^2 – 3 & \text{if } x > 4 \end{cases}f(x)=⎩⎨⎧​x+2−2x+5×2−3​if x≤1if 1<x≤4if x>4​

1. Domain:
   • The first piece x+2x + 2x+2 is valid for x≤1x \leq 1x≤1.
   • The second piece −2x+5-2x + 5−2x+5 is valid for 1<x≤41 < x \leq 41<x≤4.
   • The third piece x2−3x^2 – 3×2−3 is valid for x>4x > 4x>4.
   The domain is all real numbers because the function is defined for all x-values, but broken into different intervals. So, the domain is: (−∞,1]∪(1,4]∪(4,∞)(-\infty, 1] \cup (1, 4] \cup (4, \infty)(−∞,1]∪(1,4]∪(4,∞)
2. Range:
   • For x≤1x \leq 1x≤1, the function is x+2x + 2x+2, and as xxx decreases, yyy also decreases.
   • For 1<x≤41 < x \leq 41<x≤4, the function is −2x+5-2x + 5−2x+5, which is a linear function with a decreasing slope.
   • For x>4x > 4x>4, the function is x2−3x^2 – 3×2−3, a parabola that increases as xxx increases.
   The range depends on the output values from each piece.

• For the first piece (x+2x + 2x+2, x≤1x \leq 1x≤1): when x=1x = 1x=1, y=3y = 3y=3, and as xxx decreases, yyy decreases.
• For the second piece (−2x+5-2x + 5−2x+5, 1<x≤41 < x \leq 41<x≤4): when x=1x = 1x=1, y=3y = 3y=3, and when x=4x = 4x=4, y=−3y = -3y=−3.
• For the third piece (x2−3x^2 – 3×2−3, x>4x > 4x>4): the minimum value occurs when x=4x = 4x=4, giving y=13y = 13y=13, and the function increases as xxx increases.

Thus, the range is:(−∞,3]∪[−3,∞)(-\infty, 3] \cup [-3, \infty)(−∞,3]∪[−3,∞)

You can visualize this function using Desmos to better understand the domain and range, and verify how the function behaves across the different intervals.