Science:Math Exam Resources/Courses/MATH220/December 2010/Question 05 (b)

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MATH220 December 2010

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Question 05 (b)
Prove that the following function is bijective: $f:\mathbb {R} -\{-2\}\to \mathbb {R} -\{1\}$ defined by $f(x)={\frac {x+1}{x+2}}.$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that a function is bijective if it is one to one and onto.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

We first show our function is one to one (injective). Let $\displaystyle x,y\in \mathbb {R} -\{-2\}$ be such that $f(x)=f(y)$, that is, $\displaystyle {\frac {x+1}{x+2}}={\frac {y+1}{y+2}}$ Cross multiplying gives $\displaystyle (x+1)(y+2)=(y+1)(x+2).$ Simplifying yields $\displaystyle xy+2x+y+2=xy+2y+x+2$ and cancelling like terms shows that $\displaystyle x=y.$ To show our function is onto, suppose that $\displaystyle y\in \mathbb {R} -\{1\}$. We want to know if there is a $\displaystyle x\in \mathbb {R} -\{-2\}$ such that $f(x)=y$, that is, $\displaystyle {\frac {x+1}{x+2}}=y$ If we isolate for our x, we see that ${\begin{aligned}(x+1)&=y(x+2)\\x+1&=xy+2y\\1-2y&=xy-x\\{\frac {1-2y}{y-1}}&=x\end{aligned}}$ Notice in the last step dividing by $\displaystyle y-1$ is allowed since $\displaystyle y\neq 1$. Defining $x={\frac {1-2y}{y-1}}$ implies that $f(x)=y$. It remains to show that $x\in \mathbb {R} -\{-2\}$, that is, $x\neq -2$. Using a proof by contradiction we assume that $x=-2$, which implies ${\begin{aligned}{\frac {1-2y}{y-1}}&=-2\\1-2y&=-2y+2\\1&=2\end{aligned}}$ A contradiction! Hence, indeed, $x\in \mathbb {R} -\{-2\}$ and the proof is complete.

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