if f(x)=2x+1 and g(f(x))= 4x^2+4x+3 what is g(x) given that g(x)=ax^2+bx+c

Question

We have to find the function g(x) given the g(f(x)) composite function. 

 

 

(If f(x)=2x+1 and g(f(x))= 4x^2+4x+3 what is g(x) given that g(x)=ax^2+bx+c)

Answer 1

If you are given that g(f(x))=4x^2+4x+3, g(x)=ax^2+bx+c and that f(x)=2x+1, then in order to find a,b,c you need to substitute f(x) into the equation of g(x) anywhere that x appears in the latter function. 

 

So g(f(x))=a(2x+1)^2+b(2x+1)+c

 

Expanding that out, you get g(f(x))=a(4x^2+4x+1)+2bx+b+c=4x^2+4x+3

 

Simplifying the equation gets you: g(f(x))=4ax^2+(4a+2b)x+a+b+c=4x^2+4x+3

 

So from that you know: 4a=4, (4a+2b)=4, and a+b+c=3

 

You start with the equation with 1 unknown: 4a=4, a=1

 

Then you move onto the second one with 2 unknowns, one of which you just solved: 4a+2b=4(1)+2b=4, b=0

 

Finally, you solve the last equation: a+b+c=1+0+c=3, c=2

 

Substituting a,b,c into g(x): g(x)=x^2+2

 

As a check, substitute f(x)=2x+1 into g(x) as 'x' and see if you get the original equation given in the question back:

 

g(f(x))=(2x+1)^2+2=(4x^2+4x+1)+2=4x^2+4x+3 which is exactly what you started out with. 

Answer 2

Sarah:

 

You employ the method of substitution and then comparison of coefficients. We know the general form of g(x), hence by substituting f(x) = 2x+1 for x we get: a(2x+1)^2+b(2x+1)+c=4ax^2+4ax+1+2bx+b+c=x^2*4a+x*(4a+2b)+(a+b+c), which must be identical to 4x^2+4x+3.

Therefore 4a=4, 4a+2b=4, and a+b+c=3. From this follows a=1, b=0 and c=2. So g(x)=x^2+2.

 

Best regards,

Hanns Ewald