Math

Question Identify the hole, vertical asymptote, horizontal asymptote, and domain of the rational function f(x)=x2+9x+20x2+3x4f(x) = \frac{x^2 + 9x + 20}{x^2 + 3x - 4}.

Studdy Solution

STEP 1

1. A hole in the graph of a rational function occurs at a value of xx where both the numerator and denominator are zero, indicating a common factor that can be canceled.
2. A vertical asymptote occurs at values of xx where the denominator is zero and the numerator is not zero after canceling common factors.
3. A horizontal asymptote is determined by comparing the degrees of the numerator and denominator. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
4. The domain of a rational function consists of all real numbers except where the denominator is zero.

STEP 2

1. Factor the numerator and the denominator of the rational function.
2. Identify any common factors to determine the hole(s).
3. Identify the zeros of the denominator that are not canceled by the numerator to determine the vertical asymptote(s).
4. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator.
5. Establish the domain of the function by excluding the values that make the denominator zero.

STEP 3

Factor the numerator and the denominator of the rational function.
f(x)=x2+9x+20x2+3x4 f(x) = \frac{x^2 + 9x + 20}{x^2 + 3x - 4}
Factor the numerator:
x2+9x+20=(x+5)(x+4) x^2 + 9x + 20 = (x + 5)(x + 4)
Factor the denominator:
x2+3x4=(x+4)(x1) x^2 + 3x - 4 = (x + 4)(x - 1)

STEP 4

Identify any common factors to determine the hole(s).
The common factor in the numerator and denominator is (x+4)(x + 4).

STEP 5

Determine the value of xx for the hole by setting the common factor equal to zero.
x+4=0 x + 4 = 0
Solve for xx:
x=4 x = -4
The hole is at x=4x = -4.

STEP 6

Identify the zeros of the denominator that are not canceled by the numerator to determine the vertical asymptote(s).
The remaining factor in the denominator after canceling the common factor is (x1)(x - 1).

STEP 7

Set the remaining factor in the denominator equal to zero to find the vertical asymptote.
x1=0 x - 1 = 0
Solve for xx:
x=1 x = 1
The vertical asymptote is at x=1x = 1.

STEP 8

Determine the horizontal asymptote by comparing the degrees of the numerator and denominator.
The degrees of the numerator and denominator are both 2, and the leading coefficients are both 1.

STEP 9

Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
The horizontal asymptote is:
y=11=1 y = \frac{1}{1} = 1

STEP 10

Establish the domain of the function by excluding the values that make the denominator zero.
The denominator is zero at x=4x = -4 and x=1x = 1. Since x=4x = -4 corresponds to a hole, it is not considered a restriction for the domain.
The domain of the function is:
{xR:x1} \{ x \in \mathbb{R} : x \neq 1 \}
The hole is at x=4x = -4, the vertical asymptote is at x=1x = 1, the horizontal asymptote is at y=1y = 1, and the domain is all real numbers except x=1x = 1.

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