Introduction to Finding the Range of a Function

Determining the range of a function is a fundamental concept in mathematics, particularly essential for those working in fields such as calculus, physics, and engineering. This guide will walk you through a detailed example of how to find the range of the function f(x) frac{3x^2 2x}{5x^2 4}, using methods that make it clear and comprehensible for beginners and advanced learners alike. We'll cover how to identify the roots of the numerator, calculate the minimum and maximum points, and determine the horizontal asymptotes.

Step 1: Setting Up the Function

Let's start with the given function:

f(x) frac{3x^2 2x}{5x^2 4}

This function can be rearranged into a more manageable form:

3 - 5yx^2 2x - 4y 0

Here, we treat y as a constant. This equation is a quadratic in x. For y in mathbb{R}, the discriminant D geq 0. This is a critical step, as it ensures real roots.

Step 2: Using the Discriminant to Establish the Range

The discriminant of the quadratic equation is given by:

D b^2 - 4ac 2^2 - 4(3)(-5y - 4y) 1 - 20y^2 12y

We require:

1 - 20y^2 12y geq 0

This can be further simplified to:

20y^2 - 12y - 1 leq 0

To solve this inequality, we use the quadratic formula or factorization. The roots of the equation are:

y frac{12 pm sqrt{144 80}}{40} frac{12 pm sqrt{224}}{40} frac{3 pm sqrt{14}}{10}

Thus, the range of y is: y in left[ frac{3 - sqrt{14}}{10}, frac{3 sqrt{14}}{10} right]

Step 3: Analyzing the Critical Points and Asymptotes

To find the critical points, we calculate the derivative of f(x):

f'(x) frac{6x cdot (5x^2 4) - (3x^2 2x) cdot 1}{(5x^2 4)^2}

Simplifying, we get:

f'(x) frac{-2 cdot (5x^2 - 12x - 4)}{(5x^2 4)^2}

The critical points are where the numerator of the derivative equals zero:

5x^2 - 12x - 4 0

Using the quadratic formula, we find the roots:

x frac{12 pm sqrt{144 80}}{10} frac{12 pm sqrt{224}}{10} frac{6 pm sqrt{14}}{5}

The quadratic equation implies that:

f(x) is decreasing on (-infty, frac{6 - sqrt{14}}{5}), increasing on (frac{6 - sqrt{14}}{5}, frac{6 sqrt{14}}{5}), and decreasing on (frac{6 sqrt{14}}{5}, infty).

The minimum value occurs at x frac{6 - sqrt{14}}{5}, and the maximum value occurs at x frac{6 sqrt{14}}{5}.

The corresponding y-values are:

Minimum value: y 3 - frac{sqrt{14}}{10}

Maximum value: y 3 frac{sqrt{14}}{10}

Step 4: Determining Asymptotes and Completing the Analysis

To determine the horizontal asymptote, we find the limit as x to pm infty:

lim_{x to pm infty} f(x) frac{3}{5}

This means the horizontal asymptote is y frac{3}{5}.

Combining all the information, the range of the function f(x) is:

[3 - frac{sqrt{14}}{10}, 3 frac{sqrt{14}}{10}]

Conclusion

Understanding the range of a function is crucial for many applications, particularly in calculus and analysis. By following the steps outlined above, you can systematically determine the range of a given function, whether it be a simple or complex one. The key is to identify critical points, analyze the critical points, and use asymptotes to define the boundaries of the range.

Further Reading and Resources

For more advanced topics and in-depth analysis, consider exploring the following resources:

First Derivative Test Explained Finding the Range of a Rational Function Understanding Range in Functions