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Let $f(x) = 3x^2-2$ and $g(f(x)) = x^2 + x +1$. Find the sum of all possible values of $g(25)$.

Creeperhissboom  May 20, 2018
 #1
avatar+7155 
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To find  g(25)  , let's first find what values of  x  make  f(x) = 25

 

f(x)  =  25

                         Substitute  3x2 - 2  in for  f(x) .

3x2 - 2  =  25

                         Add  2  to both sides of the equation.

3x2  =  27

                         Divide both sides by  3 .

x2  =  9

                         Take the  ±  square root of both sides.

x  =  ±√9

 

x  =  ± 3

 

So...     f(3)  =  25     and     f(-3)  =  25

 

g( f(x) )  =  x2 + x +1

                                     Let's plug in  3  for  x .

g( f(3) )  =  32 + 3 + 1

                                     Now we can substitute  25  in for  f(3)  because we know that  f(3)  =  25 .

g( 25 )  =  32 + 3 + 1

                                     And now simplify the right side of the equation.

g( 25 )  =  13                 This is one possible value of  g( 25 ) .

 

g( f(x) )  =  x2 + x +1

                                            Now let's plug in  -3  for  x .

g( f(-3) )  =  (-3)2 + (-3) +1

                                            Substitute  25  in for  f(-3)  since  f(-3)  =  25 .

g( 25 )  =  (-3)2 + (-3) +1

                                            Simplify the right side.

g( 25 )  =  7                          This is the other possible value of  g( 25 ) .

 

The two possible values of  g( 25 )  are  13  and  7 .

 

13 + 7  =  20

hectictar  May 20, 2018

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