Find the interval(s) where $f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }$ is continuous.
A) $[ 0 , \infty )$
B) $[ - 3,3 ]$
C) $( - \infty , \infty )$
D) $[ - 2,3 ]$

Question

Quizplus · Accepted Answer

The function $f(x) = \sqrt{x^2 - 2x + 3}$ is continuous everywhere because the expression under the square root, $x^2 - 2x + 3$, is always positive for all real x. This is because the discriminant of the quadratic equation $x^2 - 2x + 3 = 0$ is negative ($b^2 - 4ac = (-2)^2 - 4(1)(3) = 4 - 12 = -8$), indicating that it has no real roots and thus the expression inside the square root is always positive. Therefore, the square root function, being applied to a positive value for all x, is continuous for all real numbers.

Find the Interval(s) Where f(x)=x2−2x+3f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }f(x)=x2−2x+3​

Find the Interval(s) Where $f ( x ) = \sqrt { x ^ { 2 } - 2 x + 3 }$