Question

Suppose a function is defined as f(x) = [(x + a)(x + b)] divided by [(x + c)(x + d)]. What are its zeros, asymptotes, and holes? Now write a rational function with numbers instead of a, b, c, and d. Give the function's zeros, asymptotes, and holes. (Note that more than one rational function can have the same zeros, asymptotes, and holes. Therefore, many answers to this question are possible.)

Answer 1

Okay... zeros of a function (in particular, a rational one, such as this) would be the values of x that would make the NUMERATOR equal to zero.

the numerator here is (x+a)(x+b)

If we set
(x+a)(x+b) = 0
What are the values of x?

Answer 2

1?

Answer 3

No....

Answer 4

-x?

Answer 5

Okay... simply put
(x+a)(x+b) = 0
only means either
x+a = 0

or

x+b = 0

Solve both these equations...

Answer 6

oh okay
x+a-a=x
x+b-b=x
Like that?

Answer 7

=0 at the right sides, and not = x

Answer 8

Oh okay

Answer 9

Okay, like this...
\[\large x+a = 0\]
subtract a from both sides
\[\large x+a\color{red}{-a}= 0 \color{red}{-a}\]
simplify
\[\large x = -a\]
tadaa\
:D

Answer 10

Thanks! :D

Answer 11

That's not all of the answers yet...

Answer 12

Just one of them.

Answer 13

Oh okay, I'm listening. Go on

Answer 14

I really have to go now... I'm sure there are others here that can help you...

The other equation is
x+b=0
Would you please solve this?

Answer 15

(Not forgetting of course, that as of now, we are only solving for the zeros of the function)

Answer 16

x+b-b=0

Answer 17

you forgot to do the same to the right-side

Answer 18

0+b-b=0?

Answer 19

uhh... no... check the way I solved the
x+a = 0
again

Answer 20

OH! x+b-b=0-b

Answer 21

Yes... continue...

Answer 22

x=-b

Answer 23

That's good.
So,
x = -b
and
x = -a
are the zeros of the function.

To find the asymptotes, this time, set the denominator
(x+c)(x+d)
to be equal to zero.

Solve
(x+c)(x+d) = 0

Answer 24

x+c-c=0-c
x=-c

x+d-d=0-d
x=-d

Answer 25

That's good :)

So, your asymptotes are
x = -c
and
x = -d


Next, the holes...
They are the values which would make BOTH the numerator and denominator equal to zero.

The zeros of the numerator are -a and -b
The zeros of the denominator are -c and -d

Do they have any in common?

Answer 26

they are negative?

Answer 27

No, I mean, does the numerator have any zeros that are also zeros of the denominator?

Answer 28

is it x?

Answer 29

No... what are the zeros of the numerator again?

Answer 30

-a and -b

Answer 31

And are any of these two also zeros of the denominator?

Answer 32

no

Answer 33

Then the function has no holes :P

Answer 34

ohh okay