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

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

The discontinuity is at x=-4.

These are correct if the function is .

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

Step-by-step explanation:

I imagine you mean 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 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.

comparing to gives you .

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 is .

So becomes  .

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.

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

Step-by-step 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.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.