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. f(x)=x/(x+4)

Verified step by step guidance

1. Identify the rational function given: $f\left(x\right)=\frac{x}{\mathrm{x+4}}$.
2. Determine the values of $x$ that make the denominator zero, since these are the points where the function is undefined and potential vertical asymptotes or holes. Set the denominator equal to zero: $x+4=0$.
3. Solve the equation $x+4=0$ to find the critical value(s) of $x$. This will give $x=-4$.
4. Check if the factor $x+4$ cancels with any factor in the numerator. Since the numerator is $x$ and does not contain $x+4$, there is no cancellation, so no hole exists at $x=-4$.
5. Conclude that the function has a vertical asymptote at $x=-4$ and no holes.

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 Q(x) = 0 are excluded from the domain and can lead to vertical asymptotes or holes.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero but the numerator is not zero at those points. These are lines x = a where the function grows without bound, indicating the graph approaches infinity or negative infinity near these x-values.

Holes in the Graph

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