Question

question 13 (1 point)
given the functions $f(x) = x^2 + 1$ and $g(x) = 3 - x$, determine an equation for the combined function $y = f(x)g(x)$.
\bigcirc a) $y = x^3 + 3x^2 - x + 3$
\bigcirc b) $y = -x^3 + 2x^2 - x + 3$
\bigcirc c) $y = -x^3 + 3x^2 - x + 3$
\bigcirc d) $y = -x^3 + 3x^2 + x - 3$

Explanation:

Step1: Multiply the two functions

We need to find \( y = f(x)g(x) \) where \( f(x)=x^{2}+1 \) and \( g(x) = 3 - x \). So we multiply \( (x^{2}+1) \) and \( (3 - x) \). Using the distributive property (FOIL for polynomials), we have:
\( (x^{2}+1)(3 - x)=x^{2}\times3+x^{2}\times(-x)+1\times3 + 1\times(-x) \)

Step2: Simplify each term

Simplify each product:
\( x^{2}\times3 = 3x^{2} \), \( x^{2}\times(-x)=-x^{3} \), \( 1\times3 = 3 \), \( 1\times(-x)=-x \)

Step3: Combine like terms

Now combine the terms: \( -x^{3}+3x^{2}-x + 3 \)

Answer:

c) \( y = -x^{3}+3x^{2}-x + 3 \)