Question

1. if $f(x)=x^2 + 3x - 8$, then determine the equation of $g(x)$ if $g(x)=-f(x + 1)+3$. 2

Explanation:

Step1: Find \( f(x + 1) \)

Substitute \( x + 1 \) into \( f(x) = x^2 + 3x - 8 \).
\[

$$\begin{align*} f(x + 1)&=(x + 1)^2 + 3(x + 1) - 8\\ &=x^2 + 2x + 1 + 3x + 3 - 8\\ &=x^2 + 5x - 4 \end{align*}$$

\]

Step2: Find \( -f(x + 1) \)

Multiply \( f(x + 1) \) by -1.
\[
-f(x + 1)=-(x^2 + 5x - 4)=-x^2 - 5x + 4
\]

Step3: Find \( g(x) \)

Add 3 to \( -f(x + 1) \).
\[

$$\begin{align*} g(x)&=-f(x + 1)+3\\ &=-x^2 - 5x + 4 + 3\\ &=-x^2 - 5x + 7 \end{align*}$$

\]

Answer:

\( g(x)=-x^2 - 5x + 7 \)