In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x+3)/x(x+4)

Verified step by step guidance

1. Identify the rational function given: $g\left(x\right)=\frac{x+3}{x(x+4)}$.
2. Determine the values of $x$ that make the denominator zero, since these are the candidates for vertical asymptotes or holes. Set the denominator equal to zero: $x(x+4)=\; 0$.
3. Solve the equation $x(x+4)=\; 0$ by setting each factor equal to zero: $x=\; 0$ and $x+\; 4\; =\; 0$ which gives $x=\; -4$.
4. Check if any factor in the numerator cancels with factors in the denominator. The numerator is $x+\; 3$, which does not share any factors with the denominator, so no factors cancel.
5. Conclude that the values $x=\; 0$ and $x=\; -4$ are vertical asymptotes, and there are no holes in the graph since no factors cancel.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the domain restrictions caused by the denominator is essential, as values that make the denominator zero are not in the domain and may lead to vertical asymptotes or holes.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator is zero but the numerator is not zero, causing the function to approach infinity or negative infinity. Identifying these points helps describe the behavior of the graph near undefined values.

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, indicating a removable discontinuity. At these x-values, the function is undefined, but the limit exists, and the graph has a gap rather than an asymptote.