Question

g(x) = ( 2x + 3 , x ≥ 4
(-2x + 3 , x ≤ 4

Answer 1

It looks like you wrote a piecewise definition but both pieces include x = 4:
g(x) = { 2x + 3, x ≥ 4
-2x + 3, x ≤ 4
That assigns two different values at x = 4 (2·4+3 = 11 and −2·4+3 = −5), so as written g is not well‑defined at x = 4.

Two ways to fix it:
- If you meant the second piece for x < 4 (strict), i.e. g(x) = {2x+3 for x ≥ 4; −2x+3 for x < 4}, then g(4) = 11.
- If you meant the first piece for x > 4, i.e. g(x) = {2x+3 for x > 4; −2x+3 for x ≤ 4}, then g(4) = −5.

Properties (for either reasonable fixing):
- Domain: all real numbers (both linear pieces cover all x).
- Left limit at 4: lim_{x→4−} g(x) = −5. Right limit at 4: lim_{x→4+} g(x) = 11. So there is a jump discontinuity at x = 4 (jump size = 16).
- Range:
- If second piece includes x = 4 (x ≤ 4), range = [−5, ∞).
- If second piece is x < 4 (strict), range = (−5, ∞).
(The right piece contributes [11, ∞), and the left piece contributes values ≥ −5.)

Tell me which inequality you intended (change one ≤ or ≥ to < or >), or say what you want done with g (evaluate, graph, check continuity, etc.), and I’ll proceed.