# How do you find the vertical asymptotes and holes of #f(x)=(x^2+4x+3)/(x+3)#?

There are no asymptotes, only a hole at

Let's factorise the numerator

Therefore,

graph{(x^2+4x+3)/(x+3) [-7.316, 6.73, -5.28, 1.743]}

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To find the vertical asymptotes and holes of the function ( f(x) = \frac{x^2 + 4x + 3}{x + 3} ):

- Identify any values of ( x ) that would make the denominator equal to zero. These values represent potential vertical asymptotes or holes in the graph.
- Determine if any of these values also make the numerator equal to zero. If so, there may be holes at those points. To find the ( y )-coordinate of the hole, evaluate the function at that point.
- If there are no common factors between the numerator and denominator that cancel out, the function has a vertical asymptote at ( x = a ), where ( a ) is the value(s) that makes the denominator zero and does not make the numerator zero.
- If there are common factors that cancel out, these points represent holes in the graph rather than vertical asymptotes. To find the ( y )-coordinate of the hole, evaluate the function at the point where the common factor was canceled out.

Apply these steps to the given function ( f(x) = \frac{x^2 + 4x + 3}{x + 3} ) to find its vertical asymptotes and holes.

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