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Question 1194408: Graph the function: f(x)= 3x+5/2x-6 indicating: asymptoms, domain and range
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52802)   (Show Source):
You can put this solution on YOUR website! .
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Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Use parentheses where required! The function you show, "3x+5/2x-6", is  .
I'm quite certain that is not what you meant. You very probably mean "(3x+5)/(2x-6)", which is  .
Graph the function using an online graphing utility if all you want to do is see the graph.
If you want to understand HOW to graph the function, the online graphing utility isn't of any use to you.
The rational function has a zero where the numerator is 0, unless the denominator is also 0 for the same value of x. For this function, the only zero is when
3x+5=0
3x=-5
x=-3/5
The rational function has an asymptote where the denominator is 0, again unless the numerator is also 0 for the same value of x. For this function, the asymptote is when
2x-6=0
2x=6
x=3
The domain of a rational function is all values of x except those that make the denominator 0. For this function, the domain is all real numbers except 3.
In general, determining the range of a rational function can be difficult. But in this case, where the numerator and denominator are both linear polynomials, it is easy.
As x gets very large positive or very large negative, the "+5" and "-6" become insignificant; the function value approaches (3x)/(2x) = 3/2. But the function value never reaches 3/2; so the range of this rational function is all real numbers except 3/2.
You can see that the function value is never 3/2 by trying to solve the equation that says the value of the function is 3/2:
FALSE, so no solution
Summary....
Asymptote: x=3
Domain: all real numbers except 3
Range: all real number except 3/2
Zeros: x=-3/5
A graph of the function and of the line y=3/2, showing x=3 excluded from the domain and y=3/2 excluded from the range....
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