Algebra-2

Let's analyze each function one by one based on the given graph:

 

A.

1. Factorization:

Factor the numerator and denominator:

The numerator .

The denominator .

So the function can be rewritten as .

After canceling the common factor (for ), we get .

 

2. Vertical asymptote:

Set the denominator , we find . This does not match the graph that the vertical asymptote is at . So this function does not represent the given graph.

 

3. Horizontal asymptote:

Since the degree of the numerator and denominator of is the same (both are of degree ), dividing the leading coefficients gives . Although the horizontal asymptote matches, due to the incorrect vertical asymptote, it's not the correct function.

 

B.

1. Factorization:

Factor the expressions:

The numerator .

The denominator .

The function becomes .

After canceling the common factor (for ), we get .

 

2. Vertical asymptote:

Set , we get , which satisfies the condition for the vertical asymptote.

 

3. Horizontal asymptote:

Since the degree of the numerator and denominator of is the same (both are of degree ), dividing the leading coefficients gives . This does not match the graph that the horizontal asymptote is at . So this function does not represent the given graph.

 

C.

1. Factorization:

Factor the expressions:

The numerator .

The denominator .

The function becomes .

After canceling the common factor

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