Recall that a function, f, is a rule that assigns an element from its domain to one (and only one) element in its range. In other words, each input corresponds to exactly one output. If a function has each output correspond to exactly one input, then that function is called one-to-one, for example, f(x) = 2(x–3).

Consider a one-to-one function, f. If f(a) = b, then another one-to-one function, f ^–1 (called f-inverse), is the inverse function of f if f ^–1(b) = a. In other words, f maps input a to output b. Then f ^–1 maps b back to a.

To derive the inverse function of a given function, f(x), write y = f(x) and solve for x. This will result in a function of y, denoted f ^–1(y). Finally, replace y with x to express the inverse function in terms of the variable, x, i.e., f ^–1(x).

Example 1

• Find the inverse of [latex]f(x)=x+2[/latex].
• Write [latex]y=x+2[/latex].
• Solve for x: [latex]x=y-2[/latex].
• Write [latex]f^{-1}(y)=y-2[/latex].
• Re-express in terms of x: [latex]f^{-1}(x)=x-2[/latex].

Example 2

• Find the inverse of [latex]f(x)=3x[/latex].
• Write [latex]y=3x[/latex].
• Solve for x: [latex]x=\frac{y}{3}[/latex].
• Write [latex]f^{-1}(y)=\frac{y}{3}[/latex].
• Re-express in terms of x: [latex]f^{-1}(x)=\frac{x}{3}[/latex].

Example 3

• Find the inverse of [latex]f(x)=\frac{x-3}{2}[/latex].
• Write [latex]y=\frac{x-3}{2}[/latex].
• Solve for x: [latex]x=2y+3[/latex].
• Write [latex]f^{-1}(y)=2y+3[/latex].
• Re-express in terms of x: [latex]f^{-1}(x)=2x+3[/latex].

Restricting the domain

Some functions have outputs that have more than input, for example, f(x) = x² has two inputs (positive and negative) for each output > 0. We can sometimes limit the domain for a function that is not one-to-one in order to make it one-to-one, for example, restrict f(x) = x² to the domain x ≥ 0. We can then derive an inverse function for a function restricted to a limited domain so that it is one-to-one just as above.

Example 4

• Find the inverse of [latex]f(x)=4(x-5)^2[/latex].
• Restrict the domain to [latex]x\ge 5[/latex] so that the function is one-to-one.
• Write [latex]y=4(x-5)^2[/latex].
• Solve for x: [latex]x=\frac{\sqrt{y}}{2}+5[/latex].
• Write [latex]f^{-1}(y)=\frac{\sqrt{y}}{2}+5[/latex].
• Re-express in terms of x: [latex]f^{-1}(x)=\frac{\sqrt{x}}{2}+5[/latex], restricted to [latex]x\ge 5[/latex].