find the range of the following functions
g(x) =x/x-2
If you mean
g(x) = x/(x-2) then the range is all real values of y except y = 1
how did you get that and how would you graph this function
as x --> ±∞
the value of x/(x-2) approaches 1, but no value of x will give us 1
try solving 1 = x/(x-2)
x-2 = x
0 = 2 , which is a contradiction, so no value of x makes x.(x-2) equal to 1
e.g. let x = 1000000
then g(x) or y = 1000000/999998 = 1.000002
let x = -1000000
then y = -1000000/-1000002 = .999998
so y = 1 is a horizontal asymptote
there is also a vertical asymptote at x=2, since x=2 makes y undefined.
the graph will be a hyperbola with major axis as y = x - 1 and centre (2,1)
To find the range of the function g(x) = x/(x-2), we need to determine the set of all possible output values of the function.
First, let's determine any restrictions on the domain of the function. Since division by zero is undefined, the denominator (x-2) must not equal zero. Therefore, x ≠ 2.
Now, let's consider the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity (x → ∞), both the numerator (x) and the denominator (x-2) increase without bound. In this case, the function g(x) approaches positive infinity (∞).
As x approaches negative infinity (x → -∞), both the numerator (x) and the denominator (x-2) decrease without bound. In this case, the function g(x) approaches negative infinity (-∞).
Based on the above analysis, we can conclude that the range of the function g(x) = x/(x-2) is all real numbers except for zero (since it is not defined in the domain). In interval notation, we can express the range as (-∞, 0) ∪ (0, ∞).