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

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2 Answers

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)

 

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