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# help

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Let \(f(x)\) be an odd function defined for all real numbers \(x,\) and let \(g(x) = f(x + 3) - 5.\) You are told that the graph of \(y = g(x)\) passes through the point \((2,-2).\) Then the graph of \(y = g(x)\) must also pass through two other points \((a,b)\) and \((c,d).\) Find \((a,b)\) and \((c,d).\)

May 8, 2019

#1
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 Since  f(x)  is an odd function defined for all real numbers  x , f( -x )  =  - f(x) ___ for all real numbers  x .  And  0  is a real number, so.... f( -0 )  =  - f(0) f(0)  =  - f(0) Here we can notice that the only way for  a = -a  to be true is if  a = 0 . Still we can add  f(0)  to both sides of the equation. 2f(0)  =  0 Divide both sides of the equation by  2 f(0)  =  0 Now let's make this match the form  f(x + 3) - 5 . Rewrite  0  as  -3 + 3 f( -3 + 3 )  =  0 Subtract  5  from both sides of the equation. f( -3 + 3 ) - 5  =  -5 Substitute  g( - 3 )  in for  f(-3  + 3) - 5 g( -3 )  =  -5 Now we can see that  y  =  g( x )  passes through the point  (-3, -5) . This makes sense because the graph of  g(x)  is shifted  3 to the left and  5  down from  f(x) .   Since  y  =  g(x)  passes through  (2, -2) , g( 2 )  =  -2 Substitute  f(2 + 3) - 5  in for  g( 2 ) f( 2 + 3 ) - 5  =  -2 f( 5 ) - 5  =  -2 Add  5  to both sides. f( 5 )  =  3 Notice that  (5, 3)  is shifted  3  to the right and  5  up from  (2, -2). Since  f( -x )  =  - f( x ) ,  f( -5 )  =  - f(5)  =  -3 f( -5 )  =  - 3 Rewrite  -5  as  -8 + 3 f( -8 + 3 )  =  -3 Subtract  5  from both sides of the equation. f( -8 + 3 ) - 5  =  -8 Substitute  g( -8 )  in for  f( -8 + 3 ) - 5 g( -8 )  =  -8 And notice  (-8, -8)  is shifted  3  to the left and  5  down from  (-5, -3) . Now we can see that  y  =  g( x )  passes through the point  (-8, -8) .
May 8, 2019
#2
+2

Very nice, hectictar.......these always confuse me....!!!!!   CPhill  May 9, 2019
#3
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Honestly, they confuse me too, haha!

hectictar  May 9, 2019