Certainly! Let's go through the steps to find the inverse of the function [tex]\( F(x) = -2x + 3 \)[/tex].
1. Substitute [tex]\( y \)[/tex] for [tex]\( F(x) \)[/tex]: We start by rewriting the function [tex]\( F(x) = -2x + 3 \)[/tex] using [tex]\( y \)[/tex] instead of [tex]\( F(x) \)[/tex].
[tex]\[
y = -2x + 3
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: To find the inverse function, we switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This means we replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] and [tex]\( x \)[/tex] with [tex]\( y \)[/tex].
[tex]\[
x = -2y + 3
\][/tex]
3. Solve for [tex]\( y \)[/tex]: Now, we need to isolate [tex]\( y \)[/tex] on one side of the equation.
[tex]\[
x - 3 = -2y
\][/tex]
To get [tex]\( y \)[/tex] by itself, we divide both sides of the equation by -2:
[tex]\[
y = \frac{x - 3}{-2}
\][/tex]
4. Simplify the expression: We simplify the expression by factoring the negative sign out of the fraction:
[tex]\[
y = \frac{3 - x}{2}
\][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( F_{\text{inv}}(x) \)[/tex]: Finally, we denote the inverse function as [tex]\( F_{\text{inv}}(x) \)[/tex].
[tex]\[
F_{\text{inv}}(x) = \frac{3 - x}{2}
\][/tex]
Hence, the inverse of [tex]\( F(x) = -2x + 3 \)[/tex] is [tex]\( F_{\text{inv}}(x) = \frac{3 - x}{2} \)[/tex].
