Multi-questions about a function

Question

 

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.

Answer 1

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

Add 2 to each  side.

x² = 2

Take square root each side.

x =±√2

f have an inverse.

Answer 2

(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.

Answer 3

(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.

 

Answer 4

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.