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Find the range of y = x/(x - 5).
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Here is another solution, which does not use the conception of limits.
The function is y = .
The range of this function is the domain of the inverse function.
Let's find the inverse function.
For it, swap x and y in the original formula to get x = and solve for y.
You have
x*(y-5) = y,
xy - 5x = y,
xy - y = 5x,
y*(x-1) = 5x,
y = .
Thus, the domain of the inverse function is the set of all real numbers except of x= 1.
Hence, the range of the original function is the same set of all real numbers except of x= 1.
Solved.
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Comment from student : The domain of the inverse function is the range of the given function. Yes?
My response : Yes. For one-to-one functions it is always so.
The given function in this exercise is one-to-one, so this statement is applicable to it.
It is practically the same as if you look into a mirror and see there your image . . .
Thank you for good question.
By the way, it is the kind of knowledge which everybody must have, who learns this topic and solves such problems.
Examples of one-to-one functions
- all linear functions y = ax + b, except of horizontal;
- all monotonic polynomial functions (like y = ax^3 + const; y = ax^5 + const and many others);
- many trigonometric functions y = tan(x) + const; y = cot(x) + const in one period domain;
- power functions y = ;
- exponential functions y = ;
- logarithmic functions y = a*log(bx) + const;
- fractional linear functions of the form y = ;
- and all other monotonic functions . . .