How do I prove that f(x)=x2−4x+3 is injective at (−∞,2) and (2,∞)

Victor Wall

Victor Wall

Answered

2022-01-12

How do I prove that f(x)=x24x+3 is injective at (,2) and (2,)

Answer & Explanation

eninsala06

eninsala06

Expert

2022-01-13Added 37 answers

Step 1
If
f(x)=f(y)
then
x24x=y24y.
Hence
x2y24(xy)=0,
which means
(xy)(x+y4)=0.
This is true if and only if x=y or x+y4=0.
Now if x,y>2, then x+y>4, so the only possibility is x=y. Same for x,y<2 (try).
vicki331g8

vicki331g8

Expert

2022-01-14Added 37 answers

Step 1
On the on the interval (2,+), we have
f(x)=f(y)(x+y)(xy)=4(xy),
suppose that xy, then we have
x+y=4, but we know that both x,y>2, a contradiction. Similar for the interval (,2).
alenahelenash

alenahelenash

Expert

2022-01-24Added 366 answers

Step 1This real valued function is obiously continuous and differentiable at every point in R.df(x)dx=2x4.Sof(x)>0,x(2,+)andf(x)<0,x(,2).So since f is properly monotonous and coninouous in those two specific intervals, is also injective, i.e. if xy then f(x)f(y) .

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