Question

What are the discontinuity and zero of the function f(x) = x^2+5x+4/x+4

Answer 1

The zeros of our function f is at x=-1.

The discontinuity is at x=-4.

These are correct if the function is
`f(x)=(x^2+5x+4)/(x+4)` .

Please let know if I did not interpret your function correctly.

Explanation:

I imagine you mean
`f(x)=(x^2+5x+4)/(x+4)` but please correct me if I'm wrong.

The zero's of a rational expression occur from it's numerator.

That is, in a fraction, the only thing that makes that fraction 0 is it's numerator.

So we need to solve
`x^2+5x+4=0` for x.

The cool thing is this one is not bad to factor since the coefficient of x^2 is 1. When the coefficient of x^2 is 1 and you have a quadratic, all you have to do is ask yourself what multiplies to be c and adds to be b.

`x^2+5x+4` comparing to
`ax^2+bx+c` gives you
`a=1,b=5,c=4`.

So we are looking for two numbers that multiply to be c and add to be b.

We are looking for two numbers that multiply to be 4 and add to be 5.

Those numbers are 1 and 4 since 1(4)=4 and 1+4=5.

The factored form of
`x^2+5x+4` is
`(x+1)(x+4)`.

So
`x^2+5x+4=0` becomes
`(x+1)(x+4)=0`.

If you have a product equals 0 then at least one of the factors is 0.

So we need to solve x+1=0 and x+4=0.

x+1=0 when x=-1 (subtracted 1 on both sides to get this).

x+4=0 when x=-4 (subtracted 4 on both sides to get this).

The zeros of our function f is at x=-1 and x=-4.

Now to find where it is discontinuous. We have to think 'oh this is a fraction and I can't divide by 0 but when is my denominator 0'. If the value for the variable is not obvious to you when the denominator is 0, just solve x+4=0.

x+4=0 when x=-4 (subtracted 4 on both sides).

So we have a contradiction at one of the zeros so x=-4 can't be a zero.

The discontinuity is at x=-4.

Answer 2

This function is discontinuous at x = 4, and has a zero at x = -1.

Explanation:

If x = -4, the denominator will be zero and thus the function will be undefined. Thus, the discontinuity is at x = -4.

To find the zero(s): Set the numerator = to 0, obtaining

x^2+5x+4 = 0. Factoring this, we get (x + 4)(x + 1) = 0. Thus, we have a zero at x = -1.

Notice that f(x) can be rewritten as

x^2 + 5x + 4 (x+4)(x+1)

f(x) = -------------------- = ---------------- = x + 1 for all x other than x = -4.

x + 4 (x+4)

This function is discontinuous at x = 4, and has a zero at x = -1.