Please help with the below. I know the answer I need help figuring out how to get the answer I am obviously doing something wrong, because I come up with a different answer that what I am supposed to. Here it is: F(X)3Xsquared-5X+1 The correct answer is 12Xsquared-70X+101, the answer I come up with is 12Xsquared-65X+76. This is a composition AKA substitution problem. Here are my notes: F(X)3Xsquared-5X+1
g(X)2X-5
(Fog)(X)F(g(x))
3 (2X-5)-5X+1
6X-15-5X+1
2(3Xsquared-5X+1)-5
6Xsquared-10X+2-5
6Xsquared-10X-3
f(x) = 3x^2-5x+1
g(x) = 2x-5
(fog)(x) = f(g) = 3g^2-5g+1
= 3(2x-5)^2 - 5(2x-5) + 1
= 3(4x^2-20x+25) - 5(2x-5) + 1
= 12x^2-60x+75-10x+25+1
= 12x^2-70x+101
Looks like you messed up on the -5(2x-5) part
To solve this composition problem, we need to correctly substitute the expression for g(x) into f(x). Let's go through the steps:
1. Start with the expression for f(x): f(x) = 3x^2 - 5x + 1.
2. Replace x with g(x) in the expression for f(x): f(g(x)) = 3(g(x))^2 - 5(g(x)) + 1.
3. Replace g(x) with its expression: f(g(x)) = 3(2x - 5)^2 - 5(2x - 5) + 1.
4. Expand and simplify the expression:
Step 1: Apply the exponent to the binomial (2x - 5)^2 = (2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x +25.
Step 2: Distribute the 3 into the binomial: 3(4x^2 - 20x + 25) = 12x^2 - 60x + 75.
Step 3: Distribute the -5 into the binomial: -5(2x - 5) = -10x + 25.
Step 4: Combine like terms: 12x^2 - 60x + 75 - 10x + 25 + 1 = 12x^2 - 70x + 101.
So, the correct answer is f(g(x)) = 12x^2 - 70x + 101, which matches the answer you provided as the correct one. The answer you obtained, 12x^2 - 65x + 76, seems to have an error in the last step of simplification. Please double-check your calculations and make sure you distribute correctly.