What is the range of the function
f(x)=1/1-x? Express your answer in interval notation. Thank you!
The function is undefined for x = 1 (x = 1 is not in the domain but otherwise all real x) but the range is from -oo < y < +oo
To find the range of a function, we need to determine all the possible values that the function can output.
For the function f(x) = 1/(1-x), let's first note that the denominator (1 - x) cannot be equal to 0 because division by zero is undefined. Therefore, we need to find the values of x for which the denominator is not zero.
Setting 1 - x ≠ 0, we solve for x:
1 - x ≠ 0
-x ≠ -1
x ≠ 1
Thus, the function is defined for all values of x except x = 1.
Now, let's consider the behavior of the function for x values close to but not equal to 1. As x approaches 1 from either side (less than 1 or greater than 1), the function approaches positive or negative infinity, respectively. However, the function never actually reaches infinity because dividing 1 by a number that gets extremely close to 0 (but never equal to 0) will result in a very large positive or negative number.
Therefore, the range of f(x) = 1/(1-x) is all real numbers except infinity, which we can express in interval notation as (-∞, ∞). Note that we do not include infinity in the interval notation because infinity is not a real number.