Domain And Range Of Rational Functions In Math.

y = 6 / (1 – x^3)^4

y’ = (72x^2) / (1 – x^3)^5

To provide a detailed answer for this question, we can start by explaining the basic concepts and techniques involved in solving problems involving rational functions.

First, it is important to understand what a rational function is. A rational function is a ratio of two polynomials, where the denominator is not equal to zero. In other words, it is a function of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.

In the case of the given problem, we have a rational function in terms of x, where y is the output or dependent variable. The denominator of the rational function is (1 – x^3)^4, which means that we need to ensure that we do not choose values of x that would make this denominator equal to zero.

Therefore, the domain of the function is all real numbers except for x values that make the denominator equal to zero. We can find these values by solving the equation (1 – x^3)^4 = 0. This equation has only one real solution, which is x = 1.

Therefore, the domain of the function is (-∞, 1) U (1, ∞).

To find the range of the function, we can use various techniques, such as graphing, differentiation or substitution. In this case, we can use substitution to find the range.

If we substitute x = sin(θ), then we can simplify the denominator as follows:

(1 – x^3)^4 = [1 – (sin(θ))^3]^4

Then, we can substitute this expression back into the original function to obtain:

y = 6 / [1 – (sin(θ))^3]^4

Now, we can observe that the denominator of this expression is always positive, since (sin(θ))^3 lies between -1 and 1, and therefore, (1 – (sin(θ))^3) is always positive.

Therefore, the range of the function is all positive real numbers (i.e., y > 0).

In summary, the domain of the function is (-∞, 1) U (1, ∞) and the range of the function is all positive real numbers.

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