Porisma

Worked solution

Reverse a linear rule and keep its domain

Let ff be a function defined by f:x↦2x+1f:x\mapsto2x+1, x∈Rx\in\mathbb R, x>0x>0. Find f−1(x)f^{-1}(x).

CJC · Functions · Example 6

Solve for the original input

Answer

f−1(x)=(x−1)/2f^{-1}(x)=(x-1)/2. Its domain is x>1x>1.

The Teacher completes the Student response space: let y=f(x)y=f(x), so f−1(y)=xf^{-1}(y)=x. From y=2x+1y=2x+1, obtain x=(y−1)/2x=(y-1)/2, hence f−1(y)=(y−1)/2f^{-1}(y)=(y-1)/2, then rename the input to obtain f−1(x)=(x−1)/2f^{-1}(x)=(x-1)/2. The printed final line gives the rule. Independently the original positive inputs produce exactly (1,∞)(1,\infty), so that is the inverse domain and its range is (0,∞)(0,\infty). The source’s earlier check f(5)=11f(5)=11 gives f−1(11)=5f^{-1}(11)=5.

How did this part go?

Solution notes

The inverse-domain statement is an explicit supplement applying the adjacent source domain/range rule; it is not invented as a separate printed ask.