To find [tex]\( f^{-1}(-2) \)[/tex], let's go through the steps of finding the inverse of the function [tex]\( f(x) = 2x - 3 \)[/tex].
1. Start with the function: [tex]\( f(x) = 2x - 3 \)[/tex].
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 2x - 3
\][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y + 3 = 2x
\][/tex]
[tex]\[
x = \frac{y + 3}{2}
\][/tex]
4. Rewrite the inverse function: [tex]\( f^{-1}(y) = \frac{y + 3}{2} \)[/tex].
5. Substitute [tex]\( y = -2 \)[/tex] into the inverse function:
[tex]\[
f^{-1}(-2) = \frac{-2 + 3}{2}
\][/tex]
[tex]\[
f^{-1}(-2) = \frac{1}{2}
\][/tex]
Thus, the value of [tex]\( f^{-1}(-2) \)[/tex] is [tex]\( \boxed{0.5} \)[/tex].
