Katie V.

asked • 10/14/16

Composite Functions & Domains

f(x)= (x+3)/(x^2-4x-5) and (x)=(square root of(2-x))
 
Algebraically determine the domain of f(x)/g(x).
 
If you could provide a detailed explanation of how this would be done it would be greatly apprecited, I have a general idea but I do not have a well enough grasp to do it confidently.
 
Also, how would the solution be checked?
 
Thank you, it is really appreciated!!
 
 

Ismael A.

That doesn't seem to be a composite fucntion the way the problem is worded. It is just one function f(x) divided by another function g(x). The domain of f(x)/g(x) are those x values for which both f(x) and g(x) are defined. In this case you have g(x)=sqrt(2-x) in the denominator so you are only allowed x values that make sqrt(2-x)>=0 (greater than or equal to zero).
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10/14/16

Michael A.

tutor
Is the second function supposed to be g(x) = √(2 -x)? If so, then f(x)/g(x)  = 
 
((x + 3)/[(x² - 4x - 5)])/(√(2 - x)) =
 
(x + 3)/[(x + 1)(x - 5)(√(2 - x))]
 
The domain of this function is all real numbers such that the denominator does not equal to 0, because division by 0 is undefined, or when the expression under the radical sign is not less than zero (because the square root of negative numbers are imaginary numbers)
 
The denominator will equal to 0 when either (x + 1) or (x - 5) = 0. It will also occur when 2 - x = 0
 
That occurs when x = -1, x = 5, and x = 2, respectively. 
 
Now, where is the expression under the radical sign negative?
 
When 2 - x < 0
 
or when x > 2
 
The domain of this function is all real numbers except x = -1 and x ≥ 2
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10/14/16

Michael A.

tutor
Correction to an earlier typo: the domain is all real numbers such that x < 2, except when x = -1 . The denominator is undefined when x ≥ 2. The thought process is still the same, though. Thank you to Kenneth S. for the correction. 
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10/14/16

1 Expert Answer

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Kenneth S. answered • 10/14/16

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If f(x)= (x+3)/(x^2-4x-5) and g(x)=(square root of(2-x)) ... [note the correction to define g]
 
then domain of g is x≤2 because obviously, if x were something like 3, you'd be wanting sq root of -1 & that's NG.
 
Domain of f is x is any real EXCEPT -1 or 5.
 
Merging these two domains & realizing that x=2 must be disallowed, we find that for f/g the domain is x<2 but also excluding -1.  I leave it to you to put that into INTERVAL NOTATION.
 
I think a tutor's earlier comments were not wholly correct.
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