Finding Range of f(x) = (2x - 3) / (x - 2): Help Needed

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SUMMARY

The range of the function f(x) = (2x - 3) / (x - 2) is all real numbers except for the value 2. This conclusion is reached by analyzing the function's behavior and recognizing that x cannot equal 2, which creates a horizontal asymptote at t = 2. The transformation t = (2x - 3) / (x - 2) leads to the realization that there are no t-values that yield f(x) = 2, confirming that 2 is excluded from the range.

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CathyLou
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Hi.

Could someone please tell me the Core 3 way of finding the range of f(x) = (2x - 3) / (x - 2).

I have no idea how to draw the graph.

Any help would be really appreciated.

Thank you.

Cathy
 
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Well, you know that x can't be 2, right?

Now, let "t" denote some arbitrary real number; we are to find out what x must be in order for f to eqal f:
[tex]t=\frac{2x-3}{x-2}\to(x-2)t=2x-3\to{x}=\frac{2t-3}{t-2}, x\neq{2}[/tex]
Thus, t=2 is an inadmissible value for the function.

Note that t=2 is a horizontal ASYMPTOTE for f(x) as x trundles off to either infinity.

Furthermore, there are no t-value such that [tex]\frac{2t-3}{t-2}=2[/tex], whereupon we have shown that the range of f is all real numbers except 2.
 
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