Question

question 12 (1 point)
given the functions $f(x) = x^2 + 1$ and $g(x) = 3 - x^2$, determine an equation for the combined function $y = f(x) - g(x)$.
\bigcirc a) $y = 2x^2 - 2$
\bigcirc b) $y = 2x^2 + 2$
\bigcirc c) $y = -2$
\bigcirc d) $y = x^2 - x - 2$

Explanation:

Step1: Substitute functions into \( y = f(x) - g(x) \)

Substitute \( f(x) = x^2 + 1 \) and \( g(x) = 3 - x^2 \) into the equation:
\( y=(x^2 + 1)-(3 - x^2) \)

Step2: Distribute the negative sign

Distribute the negative sign to the terms inside the parentheses:
\( y = x^2 + 1 - 3 + x^2 \)

Step3: Combine like terms

Combine the \( x^2 \) terms and the constant terms:
\( x^2 + x^2 = 2x^2 \) and \( 1 - 3 = -2 \), so \( y = 2x^2 - 2 \)

Answer:

a) \( y = 2x^2 - 2 \)