Based on the function f(x)=2x-4/x+4 answer the following required characteristics. What is the domain, range, positive interval, negative interval, intervals of increase and interval of decrease?

Question

If a characteristic does not apply to the function, please write not applicable.

Answer 1

I'm going to make the assumptions that the function in question is actually f(x) = (2x-4)/(x+4), in which case the answer is this:

The domain is {x ∈ R | x ≠ -4} or (-∞, -4), (-4,  ∞) which is to say that the domain consists of all rational numbers from -∞ to ∞ except for -4. The reason for this is that if x=-4 then the denominator of the function becomes -4+4 = 0 and because you can't divide by zero at x=-4 the function is undefined. In other words, this function has a vertical asymptote at x = -4.

The range is {y ∈ R | y ≠ 2} or (-∞, 2), (2, ∞) because there is no value x that will cause f(x) to equal 2. This can be determined by comparing the degrees of the numerator and denominator (the exponents on the x's). Because the degree of the numerator and denominator are both equal to each other, you can determine the horizontal asymptote is y = number multiplied by x in the numerator/number multiplied by x in the denominator. In this case that is y= 2/1 or y = 2.

The positive interval is (-∞, -4), (8, ∞) and the negative interval is (-4, 8). This can be found using a simple inequality. Because the graph has a vertical asymptote, which is effectively a hole in the graph, at -4, the parenthesis indicate the interval does not include the number -4.

The interval of increase is (-∞, -4), (-4, ∞). There is not a point on the graph where it's slope turns down, but because of that asymptote there is a point at which it is nothing happening. Therefore, the interval of decrease is not applicable.