Find The Domain, Relative Extrema Points, Asymptotes, And End Behavior Of The Function, And Graph It Without A Calculator. Note: There Are No Points Of Inflection.7. $f(x) = \frac{x^2 + 1}{2x}$

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Finding Domain, Relative Extrema Points, Asymptotes, and End Behavior of a Rational Function

Understanding the Basics of Rational Functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In this article, we will focus on finding the domain, relative extrema points, asymptotes, and end behavior of the rational function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}.

Finding the Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of a rational function, the domain is all real numbers except for the values that make the denominator equal to zero.

To find the domain of the function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}, we need to find the values of xx that make the denominator 2x2x equal to zero.

2x = 0
x = 0

Therefore, the domain of the function is all real numbers except for x=0x = 0.

Finding the Relative Extrema Points of the Function

Relative extrema points are the points on the graph of a function where the function changes from increasing to decreasing or from decreasing to increasing.

To find the relative extrema points of the function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}, we need to find the critical points of the function.

f'(x) = \frac{(2x)(2x) - (x^2 + 1)(2)}{(2x)^2}
f'(x) = \frac{4x^2 - 2x^2 - 2}{4x^2}
f'(x) = \frac{2x^2 - 2}{4x^2}
f'(x) = \frac{x^2 - 1}{2x^2}

To find the critical points, we need to set the numerator of the derivative equal to zero and solve for xx.

x^2 - 1 = 0
x^2 = 1
x = \pm 1

Therefore, the critical points of the function are x=1x = 1 and x=−1x = -1.

Finding the Asymptotes of the Function

Asymptotes are the lines that the graph of a function approaches as xx approaches infinity or negative infinity.

To find the asymptotes of the function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}, we need to examine the behavior of the function as xx approaches infinity and negative infinity.

As xx approaches infinity, the numerator of the function approaches infinity and the denominator approaches infinity. Therefore, the function approaches a horizontal asymptote.

\lim_{x \to \infty} \frac{x^2 + 1}{2x} = \lim_{x \to \infty} \frac{x}{2} = \infty

As xx approaches negative infinity, the numerator of the function approaches infinity and the denominator approaches negative infinity. Therefore, the function approaches a horizontal asymptote.

\lim_{x \to -\infty} \frac{x^2 + 1}{2x} = \lim_{x \to -\infty} \frac{x}{2} = -\infty

Therefore, the horizontal asymptote of the function is y=∞y = \infty as xx approaches infinity and y=−∞y = -\infty as xx approaches negative infinity.

Finding the End Behavior of the Function

The end behavior of a function is the behavior of the function as xx approaches infinity and negative infinity.

As xx approaches infinity, the function approaches a horizontal asymptote.

\lim_{x \to \infty} \frac{x^2 + 1}{2x} = \lim_{x \to \infty} \frac{x}{2} = \infty

As xx approaches negative infinity, the function approaches a horizontal asymptote.

\lim_{x \to -\infty} \frac{x^2 + 1}{2x} = \lim_{x \to -\infty} \frac{x}{2} = -\infty

Therefore, the end behavior of the function is that it approaches a horizontal asymptote as xx approaches infinity and negative infinity.

Graphing the Function

To graph the function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}, we need to plot the points on the graph and connect them with a smooth curve.

The graph of the function has a horizontal asymptote at y=∞y = \infty as xx approaches infinity and y=−∞y = -\infty as xx approaches negative infinity.

The graph of the function has a vertical asymptote at x=0x = 0.

The graph of the function has a relative extrema point at x=1x = 1 and x=−1x = -1.

Therefore, the graph of the function is a smooth curve that approaches a horizontal asymptote as xx approaches infinity and negative infinity, has a vertical asymptote at x=0x = 0, and has relative extrema points at x=1x = 1 and x=−1x = -1.

Conclusion

In this article, we have found the domain, relative extrema points, asymptotes, and end behavior of the rational function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}. We have also graphed the function without a calculator. The graph of the function has a horizontal asymptote at y=∞y = \infty as xx approaches infinity and y=−∞y = -\infty as xx approaches negative infinity, a vertical asymptote at x=0x = 0, and relative extrema points at x=1x = 1 and x=−1x = -1.
Q&A: Finding Domain, Relative Extrema Points, Asymptotes, and End Behavior of a Rational Function

Q: What is the domain of the function f(x)=x2+12xf(x) = \frac{x^2 + 1}{2x}?

A: The domain of the function is all real numbers except for x=0x = 0, since the denominator 2x2x cannot be equal to zero.

Q: How do you find the relative extrema points of a rational function?

A: To find the relative extrema points of a rational function, you need to find the critical points of the function by setting the numerator of the derivative equal to zero and solving for xx.

Q: What is the difference between a horizontal asymptote and a vertical asymptote?

A: A horizontal asymptote is a line that the graph of a function approaches as xx approaches infinity or negative infinity. A vertical asymptote is a line that the graph of a function approaches as xx approaches a specific value.

Q: How do you find the end behavior of a rational function?

A: To find the end behavior of a rational function, you need to examine the behavior of the function as xx approaches infinity and negative infinity.

Q: What is the significance of the relative extrema points of a function?

A: The relative extrema points of a function are the points on the graph of the function where the function changes from increasing to decreasing or from decreasing to increasing.

Q: Can a rational function have more than one horizontal asymptote?

A: No, a rational function can have at most one horizontal asymptote.

Q: Can a rational function have more than one vertical asymptote?

A: Yes, a rational function can have more than one vertical asymptote.

Q: How do you graph a rational function?

A: To graph a rational function, you need to plot the points on the graph and connect them with a smooth curve.

Q: What is the significance of the asymptotes of a function?

A: The asymptotes of a function are the lines that the graph of the function approaches as xx approaches infinity or negative infinity.

Q: Can a rational function have a hole in its graph?

A: Yes, a rational function can have a hole in its graph if there is a common factor in the numerator and denominator that is canceled out.

Q: How do you find the domain of a rational function with a hole?

A: To find the domain of a rational function with a hole, you need to exclude the value of xx that makes the denominator equal to zero.

Q: Can a rational function have a discontinuity in its graph?

A: Yes, a rational function can have a discontinuity in its graph if there is a value of xx that makes the denominator equal to zero.

Q: How do you find the domain of a rational function with a discontinuity?

A: To find the domain of a rational function with a discontinuity, you need to exclude the value of xx that makes the denominator equal to zero.

Conclusion

In this Q&A article, we have answered some common questions about finding the domain, relative extrema points, asymptotes, and end behavior of a rational function. We have also discussed some additional topics related to rational functions, such as holes and discontinuities in the graph.