can someone help with my question

Question

Let f(x)=|6-x|.

(a) State the zero(s) of the function.

(b) Which of the following is true?
A. f is an even function.
B. f is an odd function.
C. f is both even and odd.
D. f is neither even nor odd.

 

 

Okay so i worked this question out and when doing the work I got x=6 for question A which would make the answer for question B be A as well. but then when i read example C it made me question if there was another way to get another answer other than 6. Did I do this equation correctly that i listed below..

 

6-x=0

-x+6=0

-x+6-6=0-6

-x=-6

-x/-x=-6/-1

x=6

 

 

Answer 1

Given  f(x)=|6-x|

the zero(s) of the function means the roots of the functions or the value(s) of x for which the function equals to Zero(0).

 

therefore we must determine x to satisfy  f(x)=|6-x| to answer (a)

 

Thus, |6-x|=0

=> +(6-x)=0 OR -(6-x)=0

=>6-x=0       OR -6+x=0

=>6-x-6=0-6 OR -6+x+6=0+6

=>     -x=-6   OR x=6

=>-x/(-1)=-6/(-1)  OR x=6

=>x=6 OR x=6

Thus the given function has only root x=6 for f(x)=+(6-x) and f(x)=-(6-x)

 

Your solution gives only a typo at -x/-x=-6/-1 and did not solve for |6-x|=0 Yet  your answers are correct.

 

To test the roots for (b)

We need to check, f(-x)=|6-(-x)|=|6+x|

Thus, |6+x|=0
=> +(6+x)=0 OR -(6+x)=0
=>6+x=0 OR -6-x=0
=>6+x-6=0-6 OR -6-x+6=0+6
=> x=-6 OR -x=6
=>x=-6 OR -x/(-1)=6/(-1)
=>x=-6 OR x=-6

Thus the given function has only root x=-6 for f(-x)=+(6-[-x]) and f(-x)=-(6-[-x])

and f(x)≠f(-x) hence the given function is not even.

Then We need to check, -f(x)=-|6-x|

Thus

-|6-x|=0

=> -[+(6-x)]=0 OR -[-(6-x)]=0

=>-(6-x)=0 OR (6-x)=0

=>-6+x=0 OR 6-x=0

=>-6+x+6=0+6 OR 6-x-6=0-6

=>x=6 OR -x=-6

=>x=6 OR -x/(-1)=-6/(-1)

=>x=6 OR x=6

Thus the given function has only root x=6 for -f(x)=-|6-x|

But f(-x) ≠f(-x) so the given function is not Odd

 

Therefore for (b) the correct answer is option  D. f is neither even nor odd.

Answer 2

 

Hi Lindsay,

 

(a) Yes, x = 6 is the one and only zero of the function f(x).

 

(b) In order for f(x) to be an even function, f(x) needs to equal f(-x).

 

     f(-x) = |6 - (-x)| = |6 + x|

 

     As f(x) = |6 - x| ≠ |6 + x|,

 

     f(x) is not an even function.

 

     In order for f(x) to be an odd function, f(x) needs to equal -f(-x).

     -f(-x) = -|6 - (-x)| = -|6 + x|
 
     As f(x) = |6 - x| ≠ -|6 + x|,
 
     f(x) is not an odd function, either.