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Question Number 174193 by Best1 last updated on 26/Jul/22
what is the range of the inverse the relation {(x,y):y=(√(4−∣x−1∣))}???
whatistherangeoftheinversetherelation{(x,y):y=4−∣x−1∣}???
Commented by cortano1 last updated on 27/Jul/22
y=(√(4−∣x−1∣)) ; 4−∣x−1∣ ≥0 ∣x−1∣≤4⇒−3≤x≤5 ∣x−1∣=4−y^2 ; ∣x−1∣= { ((x−1; x≥1)),((1−x ; x≤1)) :} ⇒ { ((x−1=4−y^2 ; x≥1)),((1−x=4−y^2 ; x≤1)) :} ⇒ { ((x=5−y^2 ; x≥1)),((x=y^2 −3 ; x≤1)) :} ⇒f^(−1) (x)= { ((5−x^2 ; x≥1)),((x^2 −3 ; x≤1)) :}
y=4−∣x−1∣;4−∣x−1∣⩾0∣x−1∣⩽4⇒−3⩽x⩽5∣x−1∣=4−y2;∣x−1∣={x−1;x⩾11−x;x⩽1⇒{x−1=4−y2;x⩾11−x=4−y2;x⩽1⇒{x=5−y2;x⩾1x=y2−3;x⩽1⇒f−1(x)={5−x2;x⩾1x2−3;x⩽1
Commented by kaivan.ahmadi last updated on 27/Jul/22
R_f^(−1) =D_f 4−∣x−1∣≥0⇒∣x−1∣≤4⇒ −4≤x−1≤4⇒−3≤x≤5 ⇒R_f^(−1) =[−3,5]
Rf−1=Df4−∣x−1∣⩾0⇒∣x−1∣⩽4⇒−4⩽x−1⩽4⇒−3⩽x⩽5⇒Rf−1=[−3,5]

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