Let f(x) = x^2 – 81. Find f^-1(x).

Question

I need explaining this example. 

Answer 1

Hi,

The problem is to find the inverse of a function, f^-1(x).

A function and its inverse have these characteristics:

1. The domain and range are switched, (x and y are switched)
2. The graph of the inverse function is a reflection of the graph of the function across the y=x line 

 

To find the inverse of a function, algebraically,  follow these steps:

 

1.   Change f(x) to y.

• y = x² - 81

2.   Switch the y and x

• x = y² - 81

3.   Solve for y

x  +  81  = y²          (add 81 to both sides)
± √(x+81)  =  ±√(y²)     (Take the square root of both sides)
± √(x+81)   =    ± y         (Simplify then apply the Reflexive Property)
  y  = √(x+81) 

4.  Replace y with f^-1(x)

f ^-1(x) = ± √(x+81) 

 

I think this is helpful.

SN

 

Answer 2

The graph of the given function is a parabola opening upward.  The graph fails the "Horizontal Line Test" (i.e., the function is not one-to-one.  For example, f(3) = f(-3)).  So, the function f(x) = x² - 81 does not have an inverse function.  

Answer 3

f(x) = x²

1.  Replace f(x) with y

     y = x²

2.  Switch y and x

    x = y²

3. Solve for y

    y = √x

4. Replace y with f^-1(x)

   f^-1(x) = √x