SOLUTION: define the function F on [-7,-1] by F(x)=x^2+8 Determine whether F is one to one. Find the range of F

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Question 332816: define the function F on [-7,-1] by F(x)=x^2+8
Determine whether F is one to one.

Find the range of F

Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!
define the function F on [-7,-1] by F(x)=x^2+8
Determine whether F is one to one

First let's draw the graph and see if it looks like it is

one-to-one, then we'll prove whether it is or not



It looks one-to-one because it does not pass beside itself

at any point.

Let's pass some horizontal lines through it, and see if

they all pass through the graph only once.



So it looks like it passes the horizontal line test, so we 

believe that it is one-to-one.  However "looking and seeing" does not 

prove anything.  So let's prove that it is one-to-one.

PROOF:

Suppose, for contradiction, that it is not one-to one.

Then there exists two numbers a and b, both in the domain of 

F(x), which is [-7.-1], meaning that they are both negative 

since the domain contains only negative numbers, such that

          F(a) = F(b)

Then

        a² + 8 = b² + 8

therefore

       a² - b² = 0

(a - b)(a + b) = 0

a - b = 0  or  a + b = 0

    a = b  

If a = b then that contradicts the assumption that a and b
are different numbers.

The other equation a + b = 0 is not possible because a and b
are both negative numbers (since the domain [-7,1] contains 
only negative numbers, and the sum of two negative numbers is
always negative, and never 0.  Therefore we have proved that
the function is one-to-one.


The range is [f(-1), f(-7)] = [(-1)²+8, (-7)²+8] = [1+8,49+8] = [9,57]


Edwin

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