Domains of functions

F(x)=x2-7x and g(x)=5-x2
Domain of (fg)(x) is. .. . .
Domain of (f/g)(x). ..

Result

a = (Correct answer is: R) Wrong answer
b = (Correct answer is: R - { x1,x2 }) Wrong answer

Solution:

a=Ra=R
 g(x)<>0  5−x2=0 −x2+5=0 x2−5=0  a=1;b=0;c=−5 D=b2−4ac=02−4⋅1⋅(−5)=20 D>0  x1,2=−b±D2a=±202=±252 x1,2=±5=±2.2360679774998 x1=5=2.2360679774998 x2=−5=−2.2360679774998   Factored form of the equation:  (x−2.2360679774998)(x+2.2360679774998)=0 b=R−{x1,x2}=R−{2.2361,(−2.2361)} \ \\ g(x)<>0 \ \\ \ \\ 5-x^2=0 \ \\ -x^2 +5=0 \ \\ x^2 -5=0 \ \\ \ \\ a=1; b=0; c=-5 \ \\ D=b^2 - 4ac=0^2 - 4\cdot 1 \cdot (-5)=20 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ \pm \sqrt{ 20 } }{ 2 }=\dfrac{ \pm 2 \sqrt{ 5 } }{ 2 } \ \\ x_{1,2}=\pm \sqrt{ 5}=\pm 2.2360679774998 \ \\ x_{1}=\sqrt{ 5}=2.2360679774998 \ \\ x_{2}=- \sqrt{ 5}=-2.2360679774998 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -2.2360679774998) (x +2.2360679774998)=0 \ \\ b=R - \{ x_{1},x_{2} \}=R - \{ 2.2361,(-2.2361) \}

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