Question

Determine the domain and range of the function f(x):Ɍ→Ɍ such that
f(x)=(x^2)/√(x^2-4)
f(x)=(2x)/(x-2)(x+1)

Answer 1

For the function f(x) = (x^2)/√(x^2-4), the domain is all real numbers except for x values that make the denominator equal to zero. In this case, x cannot be equal to 2 or -2 since that would make the square root undefined. So the domain is all real numbers except -2 and 2.

For the range, we can consider the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, the function approaches 1, since the term (x^2)/(√(x^2-4)) simplifies to 1 when x is large. As x approaches negative infinity, the function approaches -1, since the term (x^2)/(√(x^2-4)) simplifies to -1 when x is large and negative.

Therefore, the range of the function is (-1, 1).

For the function f(x) = (2x)/((x-2)(x+1)), the domain is all real numbers except for x values that make the denominator equal to zero. In this case, x cannot be equal to 2 or -1 since that would make the denominator equal to zero. So the domain is all real numbers except -1 and 2.

To determine the range, we need to consider the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, the function approaches 0, since the numerator (2x) grows linearly while the denominator ((x-2)(x+1)) grows quadratically. As x approaches negative infinity, the function also approaches 0 for the same reasons.

Therefore, the range of the function is (-∞, ∞), meaning it can take any real value.