Question

Область значений функции у=х²−2х+2 на промежутке х∈[0; 3] совпадает с множеством:

Answer 1

У=х²-2х+2
D=4-8=-4<0  ⇒  нет пересечения с осью ОХ
а=1>0  ⇒  ветви направлены вверх  
х(верш)=2/2=1
у(верш)=у(1)=1-2+2=1  ⇒   вершина в точке (1,1)
Область значений функции у∈ [ 1,+∞) .
При х∈[0,3] область значений совпадает с множеством [1,5] , так как 
у(3)=5 , у(0)=2  (вершина находится между х=0 и х=3, причём вершина - это точка минимума функции ).
Ответ:  у∈[1,5] при х∈[0,3] .

Answer 2

Function Values on the Interval [0; 3]

The function \( y = x^2 - 2x + 2 \) on the interval \( x \in [0; 3] \) coincides with the set of values:

1. For \( x = 0 \), the value of the function is \( y = 2 \). 2. For \( x = 1 \), the value of the function is \( y = 1 \). 3. For \( x = 2 \), the value of the function is \( y = 2 \). 4. For \( x = 3 \), the value of the function is \( y = 3 \).

Therefore, the set of values for the function \( y = x^2 - 2x + 2 \) on the interval \( x \in [0; 3] \) is:

{2, 1, 2, 3}

Additional Information

The given function is a quadratic function of the form \( y = ax^2 + bx + c \) where \( a = 1 \), \( b = -2 \), and \( c = 2 \).

The discriminant of the quadratic function can be calculated using the formula \( D = b^2 - 4ac \). In this case, the discriminant is \( D = (-2)^2 - 4(1)(2) = 4 - 8 = -4 \) Since the discriminant is negative, the function does not have real roots, which means it does not intersect the x-axis.

This information provides a comprehensive understanding of the function and its behavior on the given interval.