Question

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\\(f(x)\\) is a rational function given in both standard and factored forms.

let \\(f(x) = \frac{5x^2 + 36x + 36}{8x^2 - 2x - 21} = \frac{(x + 6)(5x + 6)}{(2x + 3)(4x - 7)}\\)

find:

1. the domain in interval notation

note: use -oo for \\(-\infty\\), oo for \\(\infty\\), u for union.

1. the y intercept at the point

1. x intercepts at the point(s)

1. vertical asymptotes at \\(x =\\)

1. horizontal asymptote at \\(y =\\)

Explanation:

Step1: Find the domain of the function

Set the denominator to zero to find excluded values:

$$(2x + 3)(4x - 7) = 0 \implies x = -\frac{3}{2}, x = \frac{7}{4}$$

Step2: Find the y-intercept

Evaluate the function at \(x = 0\):

$$f(0) = \frac{5(0)^2 + 36(0) + 36}{8(0)^2 - 2(0) - 21} = -\frac{36}{21} = -\frac{12}{7}$$

Step3: Find the x-intercepts

Set the numerator to zero to find the roots:

$$(x + 6)(5x + 6) = 0 \implies x = -6, x = -\frac{6}{5}$$

Step4: Find the vertical asymptotes

Identify where the denominator is zero and numerator is non-zero:

$$x = -\frac{3}{2}, x = \frac{7}{4}$$

Step5: Find the horizontal asymptote

Compare degrees of numerator and denominator to find the limit:

$$y = \frac{5}{8}$$

Answer:

1. (-oo, -3/2) U (-3/2, 7/4) U (7/4, oo)
2. (0, -12/7)
3. (-6, 0), (-6/5, 0)
4. -3/2, 7/4
5. 5/8