# Question Video: Determining If a Function Is One-to-One Mathematics

Is the function 𝑓(𝑥) = (𝑥 − 2)/(𝑥 + 11) a one-to-one function where 𝑥 ∈ ℝ − {−11}?

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### Video Transcript

Is the function 𝑓 of 𝑥 equals 𝑥 minus two all divided by 𝑥 plus 11 a one-to-one function where 𝑥 is an element of the set of real numbers excluding negative 11?

In this question, we are given a rational function 𝑓 of 𝑥 and the domain for the function and asked to determine if the function is one to one on this domain.

We can start by recalling that we say a function is one to one if every element in the range of the function corresponds to exactly one element in the domain of the function. In other words, each element in the domain is mapped to a unique element in the range.

It is often easier to check if a function is not one to one than it is to prove that a function is one to one. We can note that if two distinct values 𝑎 and 𝑏 in the domain of 𝑓 of 𝑥 exist where 𝑓 of 𝑎 equals 𝑓 of 𝑏, then 𝑓 is not one to one on this domain. However, if we can show that for this to occur, 𝑎 must equal 𝑏, then we have shown that the function is one to one on this domain.

If we tried to directly apply this process to the function 𝑓 of 𝑥, then we would have an equation in which two unknowns appear multiple times. This would be difficult to analyze, so we instead rewrite 𝑓 of 𝑥 using algebraic division to make the equation easier to solve. We could evaluate the division using polynomial long division or equating coefficients. However, it is easier to rewrite the numerator in terms of the denominator. We have that 𝑥 minus two is equal to 𝑥 plus 11 minus 13. So we can rewrite 𝑓 of 𝑥 to have this as the numerator.

We can now divide through by the denominator to obtain that 𝑓 of 𝑥 is equal to one minus 13 over 𝑥 plus 11. We are now ready to see what happens if we set two input values of 𝑓 to have the same output value in the function. We evaluate each function by substituting 𝑥 equals 𝑎 and 𝑥 equals 𝑏 into our expression for 𝑓 of 𝑥 to obtain one minus 13 over 𝑎 plus 11 equals one minus 13 over 𝑏 plus 11.

We can now simplify the equation. First, we subtract one from both sides of the equation. Next, we divide both sides of the equation through by negative 13. Now we can take the reciprocal of both sides of the equation. This gives us that 𝑎 plus 11 is equal to 𝑏 plus 11. Finally, we can subtract 11 from both sides of the equation to see that 𝑎 equals 𝑏.

Thus, we have shown that if 𝑓 of 𝑎 equals 𝑓 of 𝑏, then 𝑎 must be equal to 𝑏. This is equivalent to saying that 𝑓 of 𝑥 is one to one. Hence, we have shown that yes, the function 𝑓 of 𝑥 on the given domain is a one-to-one function.