Let f(x) = x^(2) - 2. (Note: x squared - 2)

a) If the domain of "f" is considered to be, the set of all real numbers, does "f" have an inverse? Why?

b) If the domain of "f" is considered to be [0,8), does "f" have an inverse? Why?

c) If the domain of "f" is considered to be [-8,0) does "f" have an inverse? Why?

d) In each of the above cases in which "f" has an inverse, find the inverse and specify the domain of this inverse.
asked Feb 26, 2013 in CALCULUS

a). f(x)=x2 - 2

f-1(x) = [f(x)]-1

f-1(x) = (x2 - 2)-1

f-1(x)=1/(x2 - 2)

x2 - 2=0

x2 = 2

Take square root each side.

x =±√2

f have an inverse.

(b). f(x)=x2-2

f-1(x) = [f(x)]-1

f-1(x) = (x2 - 2)-1

f-1(x)=1/(x2 - 2)

The domain of f is considered to be (0,8)

Substitute x=0  and f(x)=8 in the equation

f-1[(0)]=(02-2)-1

f-1(0)=1/0-2

f-1[(0)]=-1/2

f have an inverse.

this is false, so the is  inverse  is exists.

(c). f(x)=x2-2

f-1(x) = [f(x)]-1

f-1(x) = (x2 - 2)-1

f-1(x)=1/(x2 - 2)

The domain of f is considered to be (-8,0)

Substitute x=-8  and f(x)=0 in the equation

f-1[(-8)]=((-8)2-2)-1

f-1(-8)=1/64-2

f-1[(-8)]=1/62

f have an inverse.

a).

The function f(x)  = y = x2 - 2.

The above function is a quadratic function.

So, the domain of the function is all real numbers.

To find the inverse of the function f (x), i.e, f -1(x), replace x with y.

x = y2 - 2.

Then, solve for y.

y2 = x + 2

y = √(x + 2).

The inverse of the function f (x), i.e, f -1(x) is √(x + 2).

Therefore, f have an inverse.