Determining the Range of Values for a Function

Understanding the range of a function is crucial for a variety of applications, from mathematical modeling to real-world problem-solving. The function in question is given by:

$f(x) frac{4ax}{x^2 - a^2}$

In this article, we will explore the steps to find the range of values that the function can take. By carefully analyzing the function, we will identify critical points and evaluate the behavior at infinity.

1. Function Analysis

The function $f(x)$ is a rational function, with a linear numerator $4ax$ and a quadratic denominator $x^2 - a^2$. This form will guide the strategy to find the range of values for $y$.

2. Finding Critical Points

To find the maximum and minimum values of the function, we will take the derivative and set it to zero. We start by differentiating $f(x)$ using the quotient rule.

The derivative is:

[ f'(x) frac{(4ax)(2x) - (x^2 - a^2)(4a)}{(x^2 - a^2)^2} frac{4ax^2 4a^3 - 4ax^2}{(x^2 - a^2)^2} frac{4a^2 - x^2}{(x^2 - a^2)^2} ]

Setting the numerator to zero, we have:

$4a^2 - x^2 0$

This simplifies to:

$x^2 4a^2$

Thus, the critical points are:

$x pm 2a$

3. Evaluating the Function at Critical Points

Evaluating $f(x)$ at the critical points:

For $x 2a$:

[ f(2a) frac{4a(2a)}{(2a)^2 - a^2} frac{8a^2}{4a^2 - a^2} frac{8a^2}{3a^2} frac{8}{3} ]

For $x -2a$:

[ f(-2a) frac{4a(-2a)}{(-2a)^2 - a^2} frac{-8a^2}{4a^2 - a^2} frac{-8a^2}{3a^2} -frac{8}{3} ]

4. Behavior at Infinity

Considering the limits as $x$ approaches positive and negative infinity, we observe:

[ lim_{x to infty} f(x) lim_{x to -infty} f(x) 0 ]

5. Summarizing the Range of Values

From the critical points and the behavior at infinity, we can summarize the range of $f(x)$:

The maximum value of $f(x)$ is $frac{8}{3}$.

The minimum value of $f(x)$ is $-frac{8}{3}$.

As $x$ approaches infinity or negative infinity, $f(x)$ approaches zero. Therefore, the range of $y f(x)$ is:

$-frac{8}{3} le y le frac{8}{3}$

Conclusion

In conclusion, the function $f(x) frac{4ax}{x^2 - a^2}$ has a range of $[-frac{8}{3}, frac{8}{3}]$. This means $y$ can take any value within this interval, including zero.

This detailed analysis helps in understanding the behavior of the function and provides a clear insight into the possible values that $y$ can take under different conditions.

Further Exploration

For a more comprehensive understanding, consider exploring how different values of $a$ affect the range and the critical points of the function. This can provide additional insights into the function's behavior and help in solving similar problems more efficiently.