Is there a hole in the graph?

Question

can someone explain this problem for me?

Determine where in the domain the function is undefined. Is there an asymptote or a hole in the graph?

1. h(x)= (x+6)/(2x+1)

2. f(x) = (x+3)/(x^2+8x+15)

Answer 1

Solve for the bottom expression in the function to find where the function is undefined. Ex: (x+5) / (x+1); Solve for x+1=0, x=negative 1. x= -1 is where function is undefined, because in a fraction, the bottom cannot equal 0.

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A hole in a function means that a specific coordinate does not exist on the line.

To find if the graph has a hole,

1) Factor anything that can be factored.

2) "cancel" any like terms that are in the top and bottom

3) Solve for x in the bottom expression.

Ex: f(x) = (x^2+2x+1) / (x^2+4x+3)

1) factor the top (x^2+2x+1) --->(x+1)(x+1)

factor the bottom (x^2+4x+3) ---> (x+3)(x+1)

And it would become [(x+1)(x+1)] / [(x+3)(x+1)]

2) "Cancel" top and bottom (x+1), and you would get (x+1) / (x+3)

3) What is the value for the bottom x if you want x+3=0?

x= negative 3, which is the hole.

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An asymptote is the vertical line (hence, x=...) that the function comes close to, but does not ever intersect.

Ex: g(x)= (x+1)/(x+5)

Solve for the bottom (x+5), x+5=0 ---> x=-5

-5 would be the asymptote. Note: there is no canceling before finding x on the bottom.

Answer 2

The function is undefined when you have a divisiton by zero.  Thus you need find the values of x in each problem that will make the denominator equal to zero.

For Problem 1, you need to solve the following equation:

    2x+1 = 0

For Problem 2, you need tosolve the equation:

   x^2+8x+15 = 0