SOLUTION: Can you please help me with the following problem: Find the domain of f(g(x)). F(x)= 4/(x-2) and g(x)=3/(x-5) I know that the domain of f(x) is (-infinity,2)U (2,infinity) an

Algebra ->  Functions -> SOLUTION: Can you please help me with the following problem: Find the domain of f(g(x)). F(x)= 4/(x-2) and g(x)=3/(x-5) I know that the domain of f(x) is (-infinity,2)U (2,infinity) an      Log On


   

Question 193570: Can you please help me with the following problem:
Find the domain of f(g(x)). F(x)= 4/(x-2) and g(x)=3/(x-5)
I know that the domain of f(x) is (-infinity,2)U (2,infinity) and g(x) is (-infinity,5)U (5,infinity)
I know that f(x) has got to be set to the number that the domain cannot use in g(x)
So that would be 5=4/(x-2)
Then multiply each side by (x-2). (x-2)(5)=(4/(x-2))(x-2) this cancel out the denominator.
Then the problem would be 5x-10=4, subtract 10 from each side to isolate the variable. 5x-10+10=4+10 => 5x=14. Then to isolate x, x would have to be divided by 5. 5x/5=14/5 => x=14/5
So would the domain be (-infinity,2)U(2,(14/5))U((14/5),5)U(5,infinity)?
please let me know if any mistakes were made
thanks!
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Find the domain of f(g(x)). F(x)= 4/(x-2) and g(x)=3/(x-5)
f[g(x)] = f[3/(x-5)] = 5/[3/(x-5) -2] = 5/[(3-2x+10)/(x-5)]
= 5/[13-2x/(x-5)/
= [5(x-5)]/[13-2x]
-----------------------
Domain: All Real numbers except x = 13/2
==============================================
Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's find the domain of g(x):


g%28x%29=3%2F%28x-5%29 Start with the given function


x-5=0 Set the denominator equal to zero


x=5 Add 5 to both sides. So plugging 5 into the denominator results in a division by zero. So we must take this value out of the domain.


So the domain of g(x) is: x can be any real number but x%3C%3E5



------------------------------------------------------------
Now let's find the domain of f(g(x))

f%28x%29=4%2F%28x-2%29 Start with the given function


f%28x%29=4%2F%283%2F%28x-5%29-2%29 Plug in g%28x%29=3%2F%28x-5%29


f%28x%29=%284%28x-5%29%29%2F%283-2%28x-5%29%29 Multiply EVERY term by the inner LCD x-5 to clear out the inner fractions.


f%28x%29=%284x-20%29%2F%283-2x%2B10%29 Distribute


f%28x%29=%284x-20%29%2F%28-2x%2B13%29 Combine like terms.


-2x%2B13=0 Set the denominator equal to zero


-2x=0-13 Subtract 13 from both sides.


-2x=-13 Combine like terms on the right side.


x=%28-13%29%2F%28-2%29 Divide both sides by -2 to isolate x.


x=13%2F2 Reduce.


So we must take x=13%2F2 out of the domain (since this value causes a division by zero)



So the domain of f(g(x)) is: x can be any real number but x%3C%3E13%2F2. In addition, we made a previous statement that x%3C%3E5


So this means that the domain of f(g(x)) in interval notation is