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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
avatar+9041 
+3

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
avatar+111466 
+2

Very nice, hectictar.......these always confuse me....!!!!!

 

 

 

cool cool cool

CPhill  May 9, 2019
 #3
avatar+9041 
+2

Honestly, they confuse me too, haha!

hectictar  May 9, 2019

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