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## Answers

let f(X) = y

y=√2+x-x²

squaring both sides

y²=2+x-x²

y² is always a positive number so it means that f(X)≥0

2+x-x²≥0

multiply the whole equation by (-)1

x²-x-2≤0 because inquality sign changes

factorise it

x²-2x+x-2≤0

x(x-2)+1(x-2)≤0

(x+1)(x-2)≤0

x=-1and +2

so there are three cases

__- infinity + -1 - 2 + +infinity__

↑-----------------↑----------------↑-----------------↑

mean domain of this function is [-1,2]

--------------------------------------------------

for range of the function

x²-x-2≤0

by completing square method

(x²-x/2 - 1/4 -2 ≤ 0

(x-1/2)² + -1/4 -2 ≤0

(x-1/2)²≤9/4

x-1/2≤+-3/2

x≤+3/2+1/2=2

and x≤-3/2+1/2=-1

so minimum value of this function is -1 and maximum value of this function is 2 so range of the fuction ∈[-1 ,2]