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.