Interval Notation?

Question

f(x) = y = 5/(x^2 - 4x - 21)

a) Write the domain in interval notation.

b) Write the range in interval notation

Answer 1

f(x) = 5/(x^2 - 4x - 21)
f(x) = 5/((x - 7)(x + 3))

Domain :
(-infinity,3) U (-3,7) U (7,+infinity)

Range :
(-infinity,0) U (0,+infinity)

Answer 2

The domain is all numbers except when the denominator is zero. This happens when
x^2 - 4 x - 21 = 0
(x - 7) (x + 3) = 0
x = 7 or x = -3
so the domain is IR - {-3, 7}, or
(-oo, -3) U (-3, 7) U (7, +oo).

The range is harder to find. The denominator is a quadratic function with positive coefficient for x^2, and therefore has a minimum value. It takes on negative values (e.g. it is -21 when x = 0), and therefore the denominator can have all positive values. Therefore the range also includes the positive numbers.

The minimum value of the denominator is found when x = 2 (the average of the zero points, -3 and 7); it is -25. Therefore the function f has maximum negative value 5/(-25) = -1/5.

Conclusion: range is (-oo, -1/5] U (0, +oo).

Answer 3

The domain refers to possible x values. The restriction here is that the term x^2-4x-21 cannot equal zero, so you can set up a quadratic equation such that x^2-4x-21=0. Completing the square gives:
x^2-4x+4-25=0
(x-2)^2=25
(x-2)=+/-5
x=-3,7
So the domain is (neg. infinity,-3)union(-3,7)union(7,infinity), where the term "union" refers to the conjunction of two intervals, which is written as an upside-down capital U.

The range consists of all the possible y-values of the function. Since the function approaches neg. infinity and infinity, at which points it is considered undefined, at the points f(-3) and f(7) respectively, the range of the function is (neg. infinity, infinity).