QuestionDetermine the horizontal asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\frac{x^{}-4}{x+}$ . We are looking for the horizontal asymptote of this function

STEP 2

The horizontal asymptote of a rational function can be found by comparing the degrees of the polynomial in the numerator and the denominator.

STEP 3

First, let's identify the degrees of the polynomials in the numerator and the denominator. The degree of a polynomial is the highest power of the variable in the polynomial.
The degree of the polynomial in the numerator is2 (from $x^{2}$ ) and the degree of the polynomial in the denominator is1 (from $x$ ).

STEP 4

Since the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, there is no horizontal asymptote.
But, if we simplify the function, we might be able to find a horizontal asymptote.

STEP 5

To simplify the function, we can factor the numerator.
$f(x)=\frac{(x-2)(x+2)}{x+2}$

STEP 6

We can cancel out the common factor $(x+2)$ in the numerator and the denominator.
$f(x)=x-2$

STEP 7

Now, the function $f(x)=x-2$ is a linear function, not a rational function. The graph of a linear function is a straight line, and it does not have a horizontal asymptote.
However, we need to remember that we cancelled out the factor $(x+2)$ , which means $x$ cannot be $-2$ . So, the function is undefined at $x=-2$ .

STEP 8

This means that the line $y=x-2$ is a horizontal asymptote for all $x$ except $x=-2$ .
So, the horizontal asymptote of the function $f(x)=\frac{x^{2}-4}{x+2}$ is $y=x-2$ for all $x$ except $x=-2$ .

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QuestionDetermine the horizontal asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\frac{x^{}-4}{x+}$ . We are looking for the horizontal asymptote of this function

STEP 2

The horizontal asymptote of a rational function can be found by comparing the degrees of the polynomial in the numerator and the denominator.

STEP 3

STEP 4

Since the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, there is no horizontal asymptote.
But, if we simplify the function, we might be able to find a horizontal asymptote.

STEP 5

To simplify the function, we can factor the numerator.
$f(x)=\frac{(x-2)(x+2)}{x+2}$

STEP 6

We can cancel out the common factor $(x+2)$ in the numerator and the denominator.
$f(x)=x-2$

STEP 7

STEP 8

This means that the line $y=x-2$ is a horizontal asymptote for all $x$ except $x=-2$ .
So, the horizontal asymptote of the function $f(x)=\frac{x^{2}-4}{x+2}$ is $y=x-2$ for all $x$ except $x=-2$ .

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QuestionDetermine the horizontal asymptote for the function f(x)=x2−4x+2f(x)=\frac{x^{2}-4}{x+2}f(x)=x+2x2−4​.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x−4x+f(x)=\frac{x^{}-4}{x+}f(x)=x+x−4​ . We are looking for the horizontal asymptote of this function

STEP 2

The horizontal asymptote of a rational function can be found by comparing the degrees of the polynomial in the numerator and the denominator.

STEP 3

STEP 4

Since the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, there is no horizontal asymptote.But, if we simplify the function, we might be able to find a horizontal asymptote.

STEP 5

To simplify the function, we can factor the numerator.f(x)=(x−2)(x+2)x+2f(x)=\frac{(x-2)(x+2)}{x+2}f(x)=x+2(x−2)(x+2)​

STEP 6

We can cancel out the common factor (x+2)(x+2)(x+2) in the numerator and the denominator.f(x)=x−2f(x)=x-2f(x)=x−2

STEP 7

STEP 8

This means that the line y=x−2y=x-2y=x−2 is a horizontal asymptote for all xxx except x=−2x=-2x=−2.So, the horizontal asymptote of the function f(x)=x2−4x+2f(x)=\frac{x^{2}-4}{x+2}f(x)=x+2x2−4​ is y=x−2y=x-2y=x−2 for all xxx except x=−2x=-2x=−2.

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QuestionDetermine the horizontal asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

Assumptions1. The function is $f(x)=\frac{x^{}-4}{x+}$ . We are looking for the horizontal asymptote of this function

Since the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, there is no horizontal asymptote.
But, if we simplify the function, we might be able to find a horizontal asymptote.

To simplify the function, we can factor the numerator.
$f(x)=\frac{(x-2)(x+2)}{x+2}$

We can cancel out the common factor $(x+2)$ in the numerator and the denominator.
$f(x)=x-2$

This means that the line $y=x-2$ is a horizontal asymptote for all $x$ except $x=-2$ .
So, the horizontal asymptote of the function $f(x)=\frac{x^{2}-4}{x+2}$ is $y=x-2$ for all $x$ except $x=-2$ .