One-to-one Functions

Section 7.9 One-to-one Functions

Not all functions have inverse functions. Here’s an example.

Example 7.9.1.

Let \(f(x) = x^2 + 2\text{.}\) I evaluated at a real number \(x\) and found \(f(x) = 6\text{.}\) Can you determine my value of \(x\text{?}\)

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We need to solve the equation \(f(x) = 6\) for \(x\text{.}\) This will prove to be a problem.

\begin{align*} f(x) &= 6\\ x^2+2 &= 6\\ x^2 &= 4\\ x &= \pm \sqrt{4} = \pm 2 \end{align*}

Thus, \(x=2\) or \(x = -2\) and it is impossible to determine which value I evaluated \(f\) at to get \(6\text{.}\) So while we have limited the possibilities to two different possible values, we cannot explicitly determine the input to the function from the output.

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The problem here is two distinct inputs \(-2 \neq 2\) result in the same output \(f(-2)=f(2)\text{.}\) Graphically, the horizontal line \(y=6\) intersects the graph at two different points corresponding to the two distinct inputs.

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More generally, if \(f(x) = y\text{,}\) then solving for \(x\) gives

\begin{equation*} x = \pm \sqrt{y- 4} \end{equation*}

and \(x\) is not a function of \(y\text{.}\) Therefore, there is no inverse function for \(f\text{.}\)

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To avoid this situation, we require that each output of a function \(f\) corresponds to exactly one input. Such functions are called one-to-one functions.

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Definition 7.9.2. One-to-one Functions.

A function \(f\) is one-to-one if whenever \(a\neq b\) are in the domain of \(f\text{,}\) then \(f(a)\neq f(b)\text{.}\) That is, distinct inputs to \(f\) result in distinct outputs.

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Example 7.9.3.

Let \(f(x) = x^2+2\text{.}\) Since \(f(-2)=f(2)\text{,}\) \(f\) is not one-to-one.

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Graphically, if distinct inputs are to give distinct outputs, a horizontal line should never pass through the graph more than once. This is called the horizontal line test.

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