Question

What are the domain and range of the piecewise function below?

mc019-1.jpg
domain: mc019-2.jpg
range: mc019-3.jpg
domain: mc019-4.jpg or mc019-5.jpg
range: mc019-6.jpg or mc019-7.jpg
domain: mc019-8.jpg
range: mc019-9.jpg
domain: mc019-10.jpg or mc019-11.jpg
range: mc019-12.jpg or mc019-13.jpg

Answer 1

I can’t see the image files you listed. Please either paste the piecewise definition here (type the formulas and interval for each piece) or re-upload the image.

Meanwhile, here’s a short checklist you can use to find domain and range of any piecewise function, plus a quick example.

How to find the domain
- For each piece, note the x-interval on which that formula is defined (including whether endpoints are included or excluded).
- The domain is the union of those intervals, and any isolated x-values given by single points.
- Watch for holes, vertical asymptotes, or excluded x-values (these remove points from the domain).
- Write the final answer using interval notation and unions.

How to find the range
- For each piece, determine the set of y-values that piece attains on its x-interval:
- A constant piece gives a single y-value (include or exclude that value depending on closed/open dots).
- A linear piece on a closed interval attains its endpoint values; on an infinite interval its y-values are unbounded in one direction.
- A quadratic/cubic/etc. you may need vertex or end behavior to find min/max on the given interval.
- Union the y-sets from all pieces. Include or exclude endpoint y-values according to whether they are attained (closed dot or included x).
- If any piece produces arbitrarily large positive or negative outputs, include +∞/−∞ appropriately (in interval notation that means an unbounded interval).

Example
f(x) = { x^2 for x ≤ 0; 2 for 0 < x < 3; −x + 1 for x ≥ 3 }
- Domain: (−∞, 0] ∪ (0, 3) ∪ [3, ∞) = (−∞, ∞) (every real x is covered).
- Range:
- x^2 on x ≤ 0 gives [0, ∞) (x^2 grows without bound as x → −∞ and minimum 0 at x=0).
- constant 2 gives {2} (but 2 is already in [0,∞)).
- −x+1 for x ≥ 3: at x=3 value = −2, as x→∞ it goes to −∞, so this piece gives (−∞, −2] .
- Union: (−∞, −2] ∪ [0, ∞) = (−∞, ∞) because these overlap via whole real line when combined with continuous pieces? (In this example the union actually covers all reals because (−∞, −2] and [0,∞) leave a gap (−2,0) but 2 covers 2 (which is inside [0,∞)); to check full coverage you'd compute precisely.)
- So final range for this particular example is (−∞, −2] ∪ [0, ∞). (Note: check whether any y in (−2,0) is produced — not in this example.)

If you post the exact piecewise definition (or upload the image), I’ll compute the precise domain and range for your function.