To reverse a one–one function, write y= f(x), solve for the old input x, then exchange the variable names. The inverse domain is the original range; its range is the original domain. For a rational rule, multiply by its denominator, collect all terms containing the old input on one side, and factor out that input. Preserve all original restrictions. For y= 2x+1x-3, x>3, rearrangement gives x(y-2)=3y+1, then x= 3y+1y-2. Since y=2+ 7x-3>2, the inverse is 3x+1x-2, x>2. Why it works: the inverse undoes the input-output mapping; substituting its formula into the original should return the starting input on the permitted domain. Check: f-1(x) means the inverse rule, not the reciprocal 1/ f(x). A formula without its correct domain does not specify the full inverse function.

To reverse a one–one function, write \(y=\mathrm f(x)\), solve for the old input \(x\), then exchange the variable names. The inverse domain is the original range; its range is the original domain.

For a rational rule, multiply by its denominator, collect all terms containing the old input on one side, and factor out that input. Preserve all original restrictions.

For \(y=\dfrac{2x+1}{x-3}\), \(x>3\), rearrangement gives \(x(y-2)=3y+1\), then \(x=\dfrac{3y+1}{y-2}\). Since \(y=2+\dfrac7{x-3}>2\), the inverse is \(\dfrac{3x+1}{x-2}\), \(x>2\).

Why it works: the inverse undoes the input-output mapping; substituting its formula into the original should return the starting input on the permitted domain.

Check: \(\mathrm f^{-1}(x)\) means the inverse rule, not the reciprocal \(1/\mathrm f(x)\). A formula without its correct domain does not specify the full inverse function.

Linear and rational inverse rules

Linear and rational inverse rules