Urgent Math Question, please show work

Question

Let f(x) = x^3-5x^2+x-7. Find a function g such that (f o g)(x) = 27x^3 + 90x^2+78x-2. Show your work

Answer 1

Since the first term of the composite function is 27x³, then g(x) will be in the form of 3x+y.

Where y = a constant that needs to be determined. Substituting 3x+y into f(x) for x,

(3x+y)³-5(3x+y)²+3x+y-7

You don't need to multiply this out to realize,

y³-5y²+y-7 must equal -2.

Using synthetic division to factor y³-5y²+y-5,

(y-5)(y²+1)=0 -->y = 5

Therefore,

g(x) = 3x+5

f(g(x)) = 27x³+90x²+78x-2

Answer 2

Since the degree of the composition is 3 this suggests that g(x) = ax +b

Then ( f ο g ) ( x) = f ( g (x) ) = ( ax +b )³ - 5 ( ax +b )² + ax +b - 7 = 27 x³ +90 x² +78 x -2.

( ax +b )³ - 5 ( ax +b )² + ax +b - 7 = 27 x³ +90 x² +78 x -2. After some expansions on the left side....

a³ x³ +( 3a²b - 5 a² ) x² + (3 a b² -10 a b +a ) x + b³ - 5b² +b - 7 = 27 x³ +90 x² +78 x -2

Now comparing coefficients of the same degree

a³ = 27 that is a =3.

3a²b - 5 a² = 90 and since a = 3 then b = 5

You can check the rest of the coefficients. If you want. But we got the a=3 and b = 5 all we needed

Hence g(x) = 3x +5