I have a question-
The range of y = 1/(x-5) is All real Numbers
True or False
I think it is false because it should be all real numbers except y shouldn't equal 0. Is that correct or don't I understand it-Please explain if I'm wrong.
Thanks
you are correct,
If you graph it, the graph will approach the x-axis , but will never reach it.
For large positive values of x, the graph will be slightly above the x-axis,
for large negative values of x, the graph will be slightly below the x-axis
if you solve
0 = 1/(x-5) and cross-multiply , you get
0 = 1 which of course is a contradiction. Thus our equaiton cannot have a solution, that is, y can never be zero.
You are correct. The statement "The range of y = 1/(x-5) is all real numbers" is false. The range of a function represents all possible values that the function can take on for a given input. In this case, the function y = 1/(x-5) has a restriction on the denominator. The denominator, x - 5, cannot equal 0, as division by zero is undefined. Therefore, x cannot be equal to 5.
To determine the range of this function, we need to examine the behavior as x approaches values close to but not equal to 5. As x gets very close to 5 from the left, y becomes infinitely large (positive infinity). Similarly, as x gets very close to 5 from the right, y becomes infinitely large but negative (negative infinity). This means that the function has no upper or lower bounds, and it approaches positive and negative infinity as x approaches 5.
Hence, the correct statement is that the range of y = 1/(x-5) is all real numbers except for zero.