Question

Inverse for h(x)=2x-4/3

Answer 1

`y=(x)/(2)+(2)/(3)`

Explanation:

• The inverse function
  `f^(-1)` of a function
  `f` must meet that if
  `f(a)=b`, then
  `f^(-1)(b)=a`.
• 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,
  `y=h(x)=2x-(4)/(3)`.
• To find the inverse function, we clear out x, as follows:
  `y=2x-(4)/(3)`⇒
  `y+(4)/(3) =2x`⇒
  `x=(y)/(2)+(2)/(3)`.
• Now that we have clear out the value of x as a function of y, we just have to replace x by y:
  `y=(x)/(2)+(2)/(3)`, which is the inverse function we have been looking for.

• To corroborate the function is correct, we can use the fact that
  `f(a)=b`, then
  `f^(-1)(b)=a`. If we take x=1, in the first equation
  `f(1)=2-(4)/(3) = (2)/(3)`. If now we replace b=2/3 in the inverse function we obtain
  `f^(-1)((2)/(3))=((2)/(3) )/(2) +(2)/(3) =1`