Answer :
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].
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].
