Did I solve these right?
1. f(x)= -3x + 1
g(x)= -4x^2 + x + 2
Find g(f(x) )
-4(-3x+1)^2+(-3x+1)+2
-4(-3x+1)^2-3x+1+2
3-3x-4(-3x+1)^2
3-3x-4+24x-36x^2
=21x-36^2-1
2.f(x)= -2x + 1
g(x)= -3x^2 -x -1
Find (g /f)(x)
-3x^2-x-1/-2x+1
3. f(x)=-x +2
g(x)= -4x2 + x + 5
Find (f o g)(x)
f(-4x^2+x+5)
-(-4x^2+x+5)+2
=4x^2-x-5+5
=4x^2-x
=21x-36^2-1
You mean
-36 x^2 + 21 x - 1
3.
-4x^2 + x + 5 + 2
= -4 x^2 + x + 7
Let's go through each problem and see if you have solved them correctly.
1. To find g(f(x)), we substitute f(x) into g(x) and simplify.
g(f(x)) = -4(-3x + 1)^2 + (-3x + 1) + 2
First, we need to expand the square.
g(f(x)) = -4(-3x)^2 + 2(-3x)(1) + (1)^2 + (-3x + 1) + 2
Simplifying further,
g(f(x)) = -4(9x^2) - 6x + 1 + (-3x + 1) + 2
g(f(x)) = -36x^2 - 6x + 1 - 3x + 1 + 2
Finally, combining like terms,
g(f(x)) = -36x^2 - 9x + 4
Your final answer should be -36x^2 - 9x + 4. It seems like you made a mistake in step -4(-3x+1)^2-3x+1+2. The squared term should be correctly expanded.
2. To find (g / f)(x), we divide g(x) by f(x).
(g / f)(x) = (-3x^2 - x - 1) / (-2x + 1)
To divide polynomials, we can use long division or synthetic division. However, the expression you wrote seems incomplete (-3x^2 - x - 1/-2x+1). Please make sure to write the correct expression for this calculation.
3. To find (f o g)(x), we substitute g(x) into f(x) and simplify.
(f o g)(x) = f(-4x^2 + x + 5)
Substitute the expression for g(x) into f(x):
(f o g)(x) = f(-4x^2 + x + 5) = -( -4x^2 + x + 5) + 2
Simplifying further,
(f o g)(x) = 4x^2 - x - 5 + 2
(f o g)(x) = 4x^2 - x - 3
Your final answer for (f o g)(x) should be 4x^2 - x - 3. It seems like you made a small mistake in step -(-4x^2 + x + 5) + 2. The negative sign should be correctly distributed inside the parentheses.
Please double-check your work and make corrections where necessary.