To find the inverse function [tex]\( f^{-1} \)[/tex] from [tex]\( y \rightarrow x \)[/tex], we need to swap the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Essentially, for every pair [tex]\((a, b)\)[/tex] in [tex]\( f(x) \rightarrow y \)[/tex], the inverse pair will be [tex]\((b, a)\)[/tex] in [tex]\( f^{-1}(y) \rightarrow x \)[/tex].
Given the function [tex]\( f \)[/tex] is defined as:
[tex]\[ f: x \rightarrow y \][/tex]
with [tex]\( f(-1) = \frac{1}{5} \)[/tex], [tex]\( f(1) = 1 \)[/tex], [tex]\( f(2) = \frac{1}{2} \)[/tex], [tex]\( f(4) = \frac{2}{5} \)[/tex], we can write it as the set of ordered pairs:
[tex]\[ f = \{(-1, \frac{1}{5}), (1, 1), (2, \frac{1}{2}), (4, \frac{2}{5})\} \][/tex]
To find the inverse function [tex]\( f^{-1} \)[/tex], we swap each pair [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex]:
[tex]\[ f^{-1} = \{(\frac{1}{5}, -1), (1, 1), (\frac{1}{2}, 2), (\frac{2}{5}, 4)\} \][/tex]
Expressing the fractions as decimal values gives:
- [tex]\( \frac{1}{5} = 0.2 \)[/tex]
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
- [tex]\( \frac{2}{5} = 0.4 \)[/tex]
Therefore, the inverse function [tex]\( f^{-1} \)[/tex] in terms of decimal representation is:
[tex]\[ f^{-1} = \{(0.2, -1), (1, 1), (0.5, 2), (0.4, 4)\} \][/tex]
This set of ordered pairs represents the inverse function [tex]\( f^{-1} \)[/tex] from [tex]\( y \rightarrow x \)[/tex].
