SOLUTION: Let f(x)=(2x^2)/(x^2-1) and g(x)=e^(x^2-4). Solve for x the equation g(x)=16. Find f(g(x)). Simplify your answer. Describe the domains of f(x) and g(x).

Algebra ->  Graphs -> SOLUTION: Let f(x)=(2x^2)/(x^2-1) and g(x)=e^(x^2-4). Solve for x the equation g(x)=16. Find f(g(x)). Simplify your answer. Describe the domains of f(x) and g(x).       Log On


   

Question 1159812: Let f(x)=(2x^2)/(x^2-1) and g(x)=e^(x^2-4).
Solve for x the equation g(x)=16.
Find f(g(x)). Simplify your answer.
Describe the domains of f(x) and g(x).
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

matrix%282%2C1%2C%22%22%2Cf%28x%29=%282x%5E2%29%2F%28x%5E2-1%29+%29
matrix%282%2C1%2C%22%22%2Cg%28x%29=e%5E%28x%5E2-4%29%29

Solve for x the equation g%28x%29=16.

matrix%282%2C1%2C%22%22%2C16=e%5E%28x%5E2-4%29%29.......take natural log of both sides
matrix%282%2C1%2C%22%22%2Cln%2816%29=ln%28e%5E%28x%5E2-4%29%29%29
matrix%282%2C1%2C%22%22%2Cln%282%5E4%29=ln%28e%5E%28x%5E2-4%29%29%29
matrix%282%2C1%2C%22%22%2C4ln%282%29=%28x%5E2-4%29+ln%28e%29%29
matrix%282%2C1%2C%22%22%2C4ln%282%29%2Fln%28e%29=%28x%5E2-4%29%29+.........ln%28e%29=1
x%5E2-4=4ln%282%29+
x%5E2=4ln%282%29+%2B4
x%5E2=4%28ln%282%29+%2B1%29
x=sqrt%284%28ln%282%29+%2B1%29%29
x=2sqrt%28ln%282%29+%2B1%29

Find f%28g%28x%29%29. Simplify your answer.

matrix%282%2C1%2C%22%22%2Cf%28g%28x%29%29=f%28e%5E%28x%5E2-4%29%29%29












Describe the domains of f%28x%29 and g%28x%29.

the domain of+f%28x%29

matrix%282%2C1%2C%22%22%2Cf%28x%29=%282x%5E2%29%2F%28x%5E2-1%29+%29
domain will be all real numbers except values of x that make denominator equal to zero, so exclude those
x%5E2-1=0+ ->x%5E2=1+->x=sqrt%281+%29=>x=1 and x=-1

domain is:
{ x element R : x%3C%3E-1 and x%3C%3E1 }


the domain of+g%28x%29
matrix%282%2C1%2C%22%22%2Cg%28x%29=e%5E%28x%5E2-4%29%29
R (all real numbers)