Question Number 148241 by 7770 last updated on 26/Jul/21

f:x→((x^2 +x−1)/(x−1)) where x≠1 find the range of the function

Answered by Olaf_Thorendsen last updated on 26/Jul/21

f(x) = ((x^2 +x−1)/(x−1)) = x+2+(1/(x−1)) lim_(x→±∞) f(x) = ±∞, limf(x) = ±∞_(x→1^± ) f′(x) = 1−(1/((x−1)^2 )) f′(x) = 0 ⇔ x = 2 f(0) = 1 and f(2) = 5 ∀x∈]0,1[∪]1,2[, f′(x) < 0 ∀x∈]−∞,0[∪]2,+∞[, f′(x) > 0 f(]−∞,0]) = ]−∞,1] f([0,1[) = [−∞,1] f(]1,2]) = [5,+∞] f([2,+∞[) = [5,+∞] • f(R\{1}) = R\]1,5[

Answered by liberty last updated on 26/Jul/21

y(x−1)=x^2 +x−1 ⇒x^2 +(1−y)x+y−1=0 ⇒Δ≥0 ⇉ (1−y)^2 −4(y−1)≥0 ⇉ (y−1)(y−5)≥0 ⇉ y≤1 ∪ y≥ 5 R_f = {y∈R∣ y≤1 ∪ y≥ 5 }