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

Verified step by step guidance

1. Start by identifying the rational function given: $h\left(x\right)=\frac{x}{x}$. Notice that the denominator is $x(x+4)$.
2. Find 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 of these values also make the numerator zero. The numerator is $x$, so it is zero at $x=\; 0$. Since $x\; =\; 0$ makes both numerator and denominator zero, this indicates a hole at $x\; =\; 0$.
5. For $x\; =\; -4$, the numerator is not zero, so this value corresponds to a vertical asymptote. Summarize: hole at $x\; =\; 0$ and vertical asymptote at $x\; =\; -4$.

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 often indicate vertical asymptotes or holes.

Vertical Asymptotes

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

Holes in the Graph

Holes occur at values of x that make both numerator and denominator zero, indicating a removable discontinuity. After factoring and simplifying the rational function, these points are excluded from the domain but do not correspond to vertical asymptotes; instead, the graph has a 'gap' or hole.