Answer:
[tex] \huge f ^{-1}(x) = -\frac{ 2}{3 - x} [/tex]
Step-by-step explanation:
to understand this
you need to know about:
given:
- [tex]f(x) = \frac{3x + 2}{x} [/tex]
to find:
- [tex] {f}^{ - 1} (x)[/tex]
tips and formulas:
inverses of common function
- [tex] + < = > - [/tex]
- [tex] \times < = > \div [/tex]
- [tex] \frac{1}{x} < = > \frac{1}{y} [/tex]
- [tex]x ^{n} < = > \sqrt[n]{y} [/tex]
let's solve:
[tex]step - 1 : define[/tex]
[tex]f(x) = \frac{3x + 2}{x} [/tex]
[tex]step - 2 : solve[/tex]
- [tex]substitute \: y \: for \: f(x) \\ y = \frac{3x + 2}{x} [/tex]
- [tex]interchange \: x \: and \: y \: and \: swap \: the \: sides \: of \: the \: equation\\ \frac{3y + 2}{y} = x[/tex]
- [tex]multiply \: both \: sides \: of \: the \: equation \: by \: y \\ \frac{3y + 2}{y} \times y = x \times y \\ 3y + 2 = xy[/tex]
- [tex]move \: the \: constant \: and \: the \: expression \: to \: the \: right \: and \: left \: hand \: sides \: and \: change \: its \: sign \: respectively \\ 3y - xy = - 2[/tex]
- [tex]factor \: out \: y \: from \: the \: expression \\ (3 - x)y = - 2[/tex]
- [tex]divide \: both \: sides \: by \: 3 - x \\ \frac{(3 - x)y}{(3 - x)} = \frac{ - 2}{3 - x} [/tex]
- [tex] y = \frac{ - 2}{3 - x} [/tex]
[tex] \therefore \: f ^{-1}(x) = -\frac{ 2}{3 - x} [/tex]
