SOLUTION: Given the functions f(x)=sqrt-3x+15 and g(x)=5/x, determine g * f(x) and find its domain. *please note that the question has a small o where I have placed an * symbol. I am not

Algebra ->  Rational-functions -> SOLUTION: Given the functions f(x)=sqrt-3x+15 and g(x)=5/x, determine g * f(x) and find its domain. *please note that the question has a small o where I have placed an * symbol. I am not      Log On


   

Question 87625: Given the functions f(x)=sqrt-3x+15 and g(x)=5/x, determine g * f(x) and find its domain.
*please note that the question has a small o where I have placed an * symbol. I am not sure what that means.
Please someone help me.
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Given:f%28x%29+=+sqrt%28-3x%2B15%29 and g%28x%29+=+5%2Fx, determine %28g%5B0%5Df%29%28x%29
What you are being asked to determine is "the composite of g and f"
This can also shown be as g%28f%28x%29%29
These are both read as "g of f of x" and it is really a function of a function.
Here's how it works:
You are given that: f%28x%29+=+sqrt%28-3x%2B15%29 so when you are asked for g(f(x)) you are being asked for: g%28sqrt%28-3x%2B15%29%29 so in the function g%28x%29+=+%285%2Fx%29 you would substitute x+=+sqrt%28-3x%2B15%29 to get:
g%28f%28x%29%29+=+5%2Fsqrt%28-3x%2B15%29
You are also being asked to find the domain which is the set of all possible values of the independent variable, in this case, that's x.
You have to look at the denominator of the composite sqrt%28-3x%2B15%29 to see that, if x were equal to 5, the the denominator would becom zero, and this is not permissable because division by zero is undefined..
So x = 5 is an excluded value of the domain.
Also you should notice that if x > 5, the denominator would become a complex number %28ai%2Bb%29 so it looks like the domain, x, can be any real number that is less than 5, or x < 5.