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+7)/(x²+4x−21)

Verified step by step guidance

1. Identify the rational function given: $h\left(x\right)=\frac{x+\; 7}{x^2+\; 4x-\; 21}$.
2. Factor the denominator to find values of $x$ that make the denominator zero, since these are potential vertical asymptotes or holes. Factor $x^2+\; 4x-\; 21$ as $(x\; +\; 7)(x\; -\; 3)$.
3. Check if any factors in the denominator also appear in the numerator. The numerator is $x\; +\; 7$, which matches one factor in the denominator.
4. Determine the vertical asymptotes by setting the factors in the denominator that do not cancel equal to zero. Here, set $x\; -\; 3\; =\; 0$ to find $x\; =\; 3$ as a vertical asymptote.
5. Identify holes by setting the canceled factor equal to zero. Since $x\; +\; 7$ cancels, set $x\; +\; 7\; =\; 0$ to find a hole at $x\; =\; -7$.

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 involves factoring the denominator and checking for zeros that do not cancel with the numerator.

Holes in the Graph

Holes occur when a factor in the denominator cancels with an identical factor in the numerator, creating a removable discontinuity. The x-value causing the zero factor corresponds to the hole, and the function is undefined at that point but can be simplified to find the hole's y-value.