Question

Did I solve this correctly?!?!?!


When looking at a rational function f of x equals the quantity x minus six times the quantity x plus three times the quantity x plus four all divided by the quantity x plus six times the quantity x minus three times the quantity x minus four , Jamal and Angie have two different thoughts. Jamal says that the function is defined at x = −6, x = 3, and x = 4. Angie says that the function is undefined at those x values. Who is correct? Justify your reasoning.

My answer:
f(x)=(x−1)(x+2)(x+4) / (x+1)(x−2)(x−4)

f(4)=(4−1)(4+2)(4+4) / (4+1)(4−2)(4−4)

Answer 1

... f(4)=(3)(6)(8) / (5)(2)(0)

f(4)=144 / 0

f(4)=Undefined

Answer 2

Is this the original function?

\(f(x) = \dfrac{(x-6)(x+3)(x+4)}{(x+6)(x-3)(x-4)}\)

Answer 3

Yep

Answer 4

Ok.
Please next time, write it that way instead of with words. It's much easier toe understand.

Answer 5

I'm Sorry abt that :/

Answer 6

Now look at the function.
Some of the things you cannot do with real numbers are:
divide by zero
take the square root of a negative number

Answer 7

In your function, there are no square roots, so forget the second thing above.
Let's only think of division by zero which is not allowed.

Answer 8

Look at the numerator. No matter what value you put in for x, the numerator is always the numerator. Even if the numerator becomes zero, there is no problem. Dividing zero by a number (ecxept by zero) is fine. Division of a number by zero is the problem.

Answer 9

Now let's look at the denominator.

\(f(x) = \dfrac{(x-6)(x+3)(x+4)}{\color{red}{(x+6)(x-3)(x-4)}}\)

If you allow the value of x to be -6, what value will the first factor x + 6 have?

\(f(x) = \dfrac{(-6-6)(-6+3)(-6+4)}{\color{red}{(-6+6)(-6-3)(-6-4)}}\)

Answer 10

When you replace x with -6, this is what you get.

\(f(x) = \dfrac{(-12)(-3)(-2)}{\color{red}{(0)(-9)(-10)}}\)

Answer 11

What is the value of the entire denominator now?
It is 0 * (-9) * (-10)
What is that equal to?

Answer 12

0?

Answer 13

Yes. We can't have zero in the denominator, so x cannot be -6.

Answer 14

I feel so stupid :P

Answer 15

Now look at the next factor in the denominator.
It is x - 3.
What value of x will make x - 3 equal zero?

Answer 16

Makes sense so far tho :)

Answer 17

What number minus 3 equals zero?

Answer 18

3?

Answer 19

Correct.
3 makes the middle factor of the denominartor equal zero, so 3 cannot be in the domain.

Answer 20

The third factor of the denominator is x - 4
If x = 4, the third factor will be zero.
That means x cannot be 4.

We see that there are three restrictions on x:
x cannot be -6
x cannot be 3
x cannot be 4
The reason is always the same. If x were any of those values, you'd have division by zero, which is not allowed.

Answer 21

In the end, Angie is correct.
The function cannot be defined at values of x that cause division by zero.

Answer 22

Ok, gtg.
Bye.

Answer 23

Ur a great helper! Thx a lot :)

Answer 24

Thanks and you're welcome.