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I have to solve for the domain of (f-g)(5)f(x) = 9 + (9/x)g(x) = (9/x)

I need help in solving (f-g)(5) and finding the domain.

I also need to solve (f/g)(9) and find the domain.

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Thomas W. | Effective Economics, Math, Finance, & Statistics TutorEffective Economics, Math, Finance, & St...
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Manipulating functions word very similarly to manipulating variables or other numbers.  We can add them, subtract, multiply, divide, etc.  You just have to keep in mind that the function is a whole "chunk" of information, not just a single number or variable in most cases.

Subtracting two functions:

f(x) - g(x) is also equal to (f-g)(x)  This literally says to subtract the g function from the f function.

f(x) = 9 + (9/x)  AND  g(x) = (9/x)  SO....

[9 + (9/x)] - [(9/x)]  Here, you can either plug in the number given in the problem or simplify.  If you notice postitive (9/x) and a negative (9/x).  They cancel each other out.  SO...

(f-g)(x) = 9  (this is true for all numbers of x, excluding zero)

(f/g)(x) is the same as f(x)/g(x)

(9 + (9/x)) / (9/x)  (since this is a complex fraction, we can flip the denominator and then multiply across)

(9 + 9/x) * (x/9) = 9x/9 + 9x/9x  (simplify)

x + 1 (plug in x = 9)

answer: 9 + 1 = 10

Concerning domain:  there are only 3 instances in which a variable within a function creates a DNE (does not exist) value.

1:  1/x = x cannot equal zero.  the denominator can never equal zero, any values for x which turn the denominator to zero is NOT part of the domain of the function.

2:  SQRT(neg. number) = evenly rooted numbers cannot be negative, such as the square root of -4 or the fourth root of -16.  (these numbers will create imaginary numbers.)  any value of x that creates a negative in this situation is not part of the domain.

3:  1/SQRT(neg. number) = this combines the previous two rules, where we cannot have a zero as denominator or a negative number within the even root.

SO....  for the function f(x), it domain of x can be all real numbers except zero:  (-infinity, 0) U (0, +infinity)

g(x) domain in all real numbers except zero.  the same as f(x)

 

Brian B. | Making Math Matter...and Easy to Understand! Making Math Matter...and Easy to Underst...
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The domain is your input, so "5" is the domain for your question.  The complete domain is all real numbers, except x cannot equal zero.  (f-g)(5)=9 because the composite function of (f-g) is (9+9/x -9/x)=9 so as long as the input is not zero the answer will be 9.  As for (f/g)=(9+9/x)(x/9)=x+1, so (f/g)(9)=9+1=10