Answer:
[tex]y=\frac{x}{2}+\frac{2}{3}[/tex]
Step-by-step explanation:
- The inverse function [tex]f^{-1}[/tex] of a function [tex]f[/tex] must meet that if [tex]f(a)=b[/tex], then [tex]f^{-1}(b)=a[/tex].
- To find the inverse function one can clear out x from the initial equation, and once obtained an expression x=f(y), replace x by y, where y=f(x).
- In this case, [tex]y=h(x)=2x-\frac{4}{3}[/tex].
- To find the inverse function, we clear out x, as follows: [tex]y=2x-\frac{4}{3}[/tex]⇒[tex]y+\frac{4}{3} =2x[/tex]⇒[tex]x=\frac{y}{2}+\frac{2}{3}[/tex].
- Now that we have clear out the value of x as a function of y, we just have to replace x by y: [tex]y=\frac{x}{2}+\frac{2}{3}[/tex], which is the inverse function we have been looking for.
- To corroborate the function is correct, we can use the fact that [tex]f(a)=b[/tex], then [tex]f^{-1}(b)=a[/tex]. If we take x=1, in the first equation [tex]f(1)=2-\frac{4}{3} = \frac{2}{3}[/tex]. If now we replace b=2/3 in the inverse function we obtain [tex]f^{-1}(\frac{2}{3})=\frac{\frac{2}{3} }{2} +\frac{2}{3} =1[/tex]
