Raj asked in Science & MathematicsMathematics · 1 decade ago

Hi need help on this long question...?

Consider the function f(x)=(x^2+4)/(x^2-4)

What is the domain of definition of this function?

for what values of x is this function :-

1) Differentiable?

2)Continuous?

3)Increasing?

4)Decreasing?

hi im a foreign student, so Im not understand the question. could anyone help me? It would be helpful the step by step is shown

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  • 1 decade ago
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    The domain of a function is, strictly speaking, the set of input values where the function is defined.

    In this case, the function is not defined where it requires you to divide by zero, thus, it is defined everywhere except at x = 2 and x = -2, because these are the solution values to (x^2 - 4) = 0.

    So the domain of definition is the set of real numbers, without -2 or 2.

    1) Differentiable means that the derivative is defined for some values of x (the curve is smooth and continuous over those values of x). As far as I know, rational functions (functions expressed as the quotient (a fraction) of two polynomials) are differentiable everywhere they are defined.

    2) The definition says that a function is continuous at a when "the limit of f(x) as x approaches a equals f(a)". An easier test says a function is continuous where there aren't any holes or jumps in the graph.

    3) increasing: one test says that a function is increasing everywhere the derivative is positive. It's easy to see on a graph that for an increasing function, the curve is moving upward as you move right.

    4) decreasing: This is exactly the opposite of increasing: as you move right, the curve drops instead of rising.

    Now, the described function f(x) = (x² + 4)/(x² - 4) is a rational function, so it's differentiable everywhere except where the denominator is zero. Anywhere a function is differentiable, it is also continuous.

    The function has asymptotes at 2 and -2, and approaches 1 as x approaches either positive or negative infinity. Thus, any changes in whether the function is increasing or decreasing occur either where the derivative is undefined or at zero. [g(x)f'(x) - f(x)g'(x)]/g(x)², I believe, was the formula for the derivative of f(x)/g(x)

    f'(x) = [(x² - 4)(2x) - (x² + 4)(2x)]/[(x² - 4)²]

    f'(x)= -16x/[x^4 - 8x² + 16]

    So the derivative is 0 at 0 and undefined at plus-or-minus 2. These are the values where the function might change direction, so we need to check each section.

    At values less than -2, it's a negative numerator and a positive denominator; the derivative is negative, so the function is decreasing from -∞ to -2.

    For values between -2 and 0, both the numerator and denominator are negative, so the derivative is positive; the function is increasing.

    Passing zero, the numerator changes signs, but the denominator doesn't; the function changes direction, so it's decreasing from 0 to 2.

    For values greater than 2, both the numerator and denominator are positive, so the derivative is positive, and the function is increasing.

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