Math

QuestionAnalyze the function R(x)=11x+114x+8R(x)=\frac{11 x+11}{4 x+8}: find domain, vertical and horizontal asymptotes, and the correct graph.

Studdy Solution

STEP 1

Assumptions1. The function is given as R(x)=11x+114x+8R(x)=\frac{11x+11}{4x+8} . We are asked to find the domain, vertical asymptote, horizontal or oblique asymptote and the correct graph of the function.

STEP 2

First, let's find the domain of the function. The domain of a function is the set of all real numbers for which the function is defined. For a rational function like ours, the function is undefined when the denominator is zero.
So, we set the denominator equal to zero and solve for xx.
4x+8=04x+8=0

STEP 3

olve the equation for xx.
x+8=0x=2x+8=0 \Rightarrow x=-2

STEP 4

So, the function is undefined at x=2x=-2. Therefore, the domain of the function is all real numbers except -2. So, the correct answer for the domain isC. {xx2}\{x \mid x \neq-2\}

STEP 5

Next, let's find the vertical asymptote of the function. A vertical asymptote of a function is a vertical line x=ax=a where the function approaches infinity as xx approaches aa. For a rational function, the vertical asymptote is where the denominator is zero and the numerator is not zero.
From our previous step, we know that the denominator is zero at x=2x=-2. So, x=2x=-2 is the vertical asymptote.

STEP 6

So, the correct answer for the vertical asymptote isA. x=2x=-2

STEP 7

Now, let's find the horizontal or oblique asymptote of the function. A horizontal asymptote of a function is a horizontal line y=by=b where the function approaches bb as xx approaches infinity. For a rational function, if the degree of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.
Here, the degree of the numerator and the denominator are the same (both are1). So, the horizontal asymptote is the ratio of the leading coefficients, which is 114\frac{11}{4}.

STEP 8

So, the correct answer for the horizontal asymptote isA. y=114y=\frac{11}{4}

STEP 9

Finally, let's identify the correct graph of the function. The graph of the function should reflect the domain, vertical asymptote, and horizontal asymptote we found. The graph should be undefined at x=2x=-2 (represented by a hole or break in the graph), should approach the line x=2x=-2 as xx gets close to -2 (represented by the graph getting very close to the line x=2x=-2 but never touching or crossing it), and should approach the line y=114y=\frac{11}{4} as xx goes to infinity or negative infinity (represented by the graph getting very close to the line y=114y=\frac{11}{4} as it goes far to the left or right).
The correct graph is the one that meets all these criteria.

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