+166 Let $f(x) = 3x^2-2$ and $g(f(x)) = x^2 + x +1$. Find the sum of all possible values of $g(25)$.
+9465 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
