Question 1196261
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It appears to be asking what values of a,b,c will make f(x) have a range of the set of all real numbers.


If the function is {{{f(x) = (x-a)(x-b)(x-c)}}}, then it doesn't matter what a,b,c are. The range of this is the set of all real numbers. Why? Because we have a cubic. All cubics can produce any real number output for some input(s).


If instead the function is {{{f(x) = ((x-a)(x-b))/(x-c)}}}, then check out these links to see how to do a formal proof
<a href = "https://www.youtube.com/watch?v=YDukXrhjpC0">https://www.youtube.com/watch?v=YDukXrhjpC0</a>
and
<a href = "https://byjus.com/question-answer/for-real-x-the-function-frac-x-a-x-b-x-c-will-assume-all-5/">https://byjus.com/question-answer/for-real-x-the-function-frac-x-a-x-b-x-c-will-assume-all-5/</a>


The conclusion is that c is between 'a' and b
So either a < c < b or b < c < a
Or <font color=red>a > c > b</font> or b > c > a
All that matters is that c is in the middle. 


Answer: <font color=red>a > c > b</font>


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Here's another approach you could take using numeric examples


Let's say a = 1, b = 2, c = 3 which fits the a < b < c scenario
Then graph out {{{f(x) = ((x-a)(x-b))/(x-c)}}}, i.e. {{{f(x) = ((x-1)(x-2))/(x-3)}}} in this case, and you would get this 
<a href = "https://www.desmos.com/calculator/umoamx9mlz">https://www.desmos.com/calculator/umoamx9mlz</a>
Desmos is a free graphing calculator.
Notice the gap of y outputs that cannot be reached. Eg: y = 2 is in that gap
The range of this function is not the set of all real numbers.
So a < b < c is clearly not the answer. 


Now try something like a = 3, b = 2, c = 1 which fits the a > b > c scenario
So we'll be graphing {{{f(x) = ((x-3)(x-2))/(x-1)}}} to get this
<a href = "https://www.desmos.com/calculator/frzlcau2hc">https://www.desmos.com/calculator/frzlcau2hc</a>
Once again we have a gap of y values we can't reach. Eg: we can't get y = -2 as an output
So we can rule out the case of a > b > c


This trial and error process is continued until you find that <font color=red>a > c > b</font> is the answer
Let's say a = 3, b = 1, c = 2
So we want to graph {{{f(x) = ((x-3)(x-1))/(x-2)}}} to get this
<a href = "https://www.desmos.com/calculator/puw4wq8rh2">https://www.desmos.com/calculator/puw4wq8rh2</a>
There aren't any gaps for the y outputs now. 
Any output is possible.
Therefore, the range of this is the set of all real numbers.
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