SOLUTION: Given the functions 𝑓(𝑥) = √25 − 𝑥 , 𝑔(𝑥) = 𝑥^2+ 9 , ℎ(𝑥) = √𝑥 − 5 a) Obtain the domains of each given function. b) Calculate the domain and rul

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Given the functions 𝑓(𝑥) = √25 − 𝑥 , 𝑔(𝑥) = 𝑥^2+ 9 , ℎ(𝑥) = √𝑥 − 5 a) Obtain the domains of each given function. b) Calculate the domain and rul      Log On


   

Question 1194818: Given the functions 𝑓(𝑥) = √25 − 𝑥 , 𝑔(𝑥) = 𝑥^2+ 9 , ℎ(𝑥) = √𝑥 − 5
a) Obtain the domains of each given function.
b) Calculate the domain and rule of ℎ ∘ 𝑔 .
c) Calculate the domain of 𝑔 ∘ 𝑓 .
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Given the functions:
f%28x%29+=+sqrt%2825+-+x%29++
g%28x%29+=+x%5E2%2B+9+
h%28x%29+=+sqrt%28x+-5%29

a) Obtain the domains of each given function.
f%28x%29+=+sqrt%2825+-+x%29++
since given sqrt%2825+-+x%29, means 25+-+xmust be zero or greater than zero
25+-+x%3E=0
25+%3E=x
domain: { x element R : x%3C=25 }
g%28x%29+=+x%5E2%2B+9+
domain: R (all real numbers)
h%28x%29+=+sqrt%28x+-5%29
since given sqrt%28x+-+5%29,
x+-+5%3E=0
x+%3E=5
domain: { x element R : x+%3E=5 }


b) Calculate the domain and rule of ℎ ∘ 𝑔 .
ℎ ∘𝑔 =h%28g%28x%29%29=h%28x%5E2%2B+9%29=+sqrt%28x%5E2%2B+9+-5%29=sqrt%28x%5E2%2B+4%29
since x%5E2 is always positive no matter what x value is,
domain: R (all real numbers)


c) Calculate the domain of 𝑔 ∘ 𝑓 .
𝑔 ∘𝑓 =g%28f%28x%29%29=g%28sqrt%2825+-+x%29+%29=%28sqrt%2825+-+x%29+%29%5E2%2B+9=25-x%2B9=34-x
domain: R (all real numbers)