Question

f(x)={
−(x−4)
2
+6
−2
​

for x

=1
1
1
​

for x=1
1
1

Answer 1

I’m not sure I parsed the piecewise definition correctly. Do you mean

f(x) = { −(x − 4)^2 + 6 for x ≠ 1
1 for x = 1 } ?

If so, here are a few quick facts:

- Limit as x → 1 of f(x): since for x ≠ 1 the formula is the polynomial −(x − 4)^2 + 6, evaluate at x = 1:
−(1 − 4)^2 + 6 = −9 + 6 = −3. So lim_{x→1} f(x) = −3.

- Value at 1: f(1) = 1 (given).

- Continuity at 1: f is not continuous at x = 1 because lim_{x→1} f(x) = −3 ≠ f(1) = 1. The discontinuity is a removable one in the sense that the two-sided limit exists (−3) but the function’s value differs.

- Elsewhere: the formula −(x − 4)^2 + 6 is a downward-opening parabola with vertex at (4,6), so the function equals that continuous parabola for all x ≠ 1.

If I misread the piecewise definition, or if you want the limit, continuity, graph, or to choose a value to make f continuous, tell me the intended formula and I’ll proceed.