SOLUTION: given f(x)=√2x-1 and g(x)= x^2+1,x grater then or equal to 0 1.without finding the formula for f^-1,find f^-1(5) 2.find f^-1(x) and g^-1(x) 3.give range of f^-1(x) and

Algebra ->  Functions -> SOLUTION: given f(x)=√2x-1 and g(x)= x^2+1,x grater then or equal to 0 1.without finding the formula for f^-1,find f^-1(5) 2.find f^-1(x) and g^-1(x) 3.give range of f^-1(x) and      Log On


   

Question 1084544: given f(x)=√2x-1 and g(x)= x^2+1,x grater then or equal to 0
1.without finding the formula for f^-1,find f^-1(5)
2.find f^-1(x) and g^-1(x)
3.give range of f^-1(x) and give the domain of g^-1(x)
4.express (g o f)(x) in simplified form
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given f%28x%29=sqrt%282x-1%29+and g%28x%29=+x%5E2%2B1, where x+%3E=0
1.without finding the formula for f%5E-1,find+f%5E-1%285%29
Determining the value of +f%5E-1%285%29 is equivalent to determining the x
value when +f%28x%29=5
5=sqrt%282x-1%29+
5%5E2=%28sqrt%282x-1%29+%29%5E2
25=2x-1
25%2B1=2x
26%2F2=x
x=13
+f%5E-1%285%29=13

2.find f%5E-1%28x%29 and g%5E-1%28x%29
f%28x%29=sqrt%282x-1%29+
y=sqrt%282x-1%29+.............swap x and y
x=sqrt%282y-1%29+.......solve for y
x%5E2=%28sqrt%282y-1%29%29%5E2+
x%5E2=2y-1+
x%5E2%2B1=2y+
%281%2F2%29%28x%5E2%2B1%29=y+

f%5E-1%28x%29=%281%2F2%29%28x%5E2%2B1%29

g%28x%29=+x%5E2%2B1
y=+x%5E2%2B1
x=+y%5E2%2B1
x-1=+y%5E2
y=sqrt%28x-1%29
g%5E-1%28x%29=sqrt%28x-1%29


3.give range of f^-1(x) and give the domain of g^-1(x)
f%5E-1%28x%29=x%5E2%2B1
domain: R (all real numbers)
(-infinity,infinity)
range: { y element R : y%3E=1 }
[1,infinity)


4.express %28g+o+f%29%28x%29 in simplified form
I usually convert %28+f+o+g%29%28x%29 to the more intuitive +f+%28g%28x%29%29 :
+f+%28g%28x%29%29=f%28+x%5E2%2B1%29
f%28+x%5E2%2B1%29=sqrt%282%28+x%5E2%2B1%29-1%29+
f%28+x%5E2%2B1%29=sqrt%282x%5E2%2B2-1%29+
f%28+x%5E2%2B1%29=sqrt%282x%5E2%2B1%29+