Find a portion of the domain where the function is one-to-one and find an inverse function.

Question

The function ƒ(x) = −x² − 9 is not one-to-one. Find a portion of the domain where the function is one-to-one and find an inverse function.

Answer 1

Finding an Inverse Function :
1. Use the Horizontal Line Test to decide whether f has an inverse function.
2. In the equation for f(x), replace f(x) by y.
3. Interchange the roles of x and y and solve for y.
4. Replace y by f ^{- 1}(x) in the new equation.
5. Verify that f and f ^{- 1} are inverse functions of each other by showing that the domain of f is equal to the range of f ^{- 1}, the range of f is equal to the domain of f ^{- 1}, and f ( f ^{- 1}(x) ) = x and f ^{- 1}( f(x) ) = x.

The function is f(x) = - x² - 9.

(1). Applying the Horizontal Line Test :

The graph of the function given by f(x) = - x² - 9 is shown in the above Figure. Because it is possible to find a horizontal line that intersects the graph of f at more than one point, you can conclude that f is not a one-to-one function and does not have an inverse function.

Second metohod to find f is one - to - one finction : Table method:

Make the table of values to find ordered pairs that satisfy the equation.

Choose values for x and find the corresponding values for y.

x	f(x) = - x2 - 9	(x, y)
- 3	f(x) = - (- 3)2 - 9 = - 18	(- 3, - 18)
- 1	f(x) = - (- 1)2 - 9 = - 10	(- 1, - 10)
0	f(x) = - (0)2 - 9 = - 9	(0, - 9)
1	f(x) = - (1)2 - 9 = - 10	(1, - 10)
3	f(x) = - (3)2 - 9 = - 18	(- 3, - 18)

x	y
- 18	- 3
- 10	- 1
- 9	0
- 10	1
- 18	- 3

 

 

 

 

 

 

 

The table on the left is a table of values for f(x) = - x² - 9. The table of values on the right is made up by interchanging the columns of the first table. The table on the right does not represent a function because the input x = - 18 is matched with two different outputs: y = - 3 and y = 3. So, f(x) = - x² - 9 is not one-to-one and does not have an inverse function.

Algebrecally :

The original function f(x) = - x² - 9.

Replace f(x) by y.

y = - x² - 9

Interchange x and y.

x = - y² - 9

Solve for y.

x  + 9 = - y²

- x  - 9 = y²

y = ± √(- x  - 9).

Replace y by f ^{- 1}(x).

The ± indicates that to a given value of x there correspond two values of y. So, y is not a function of x.

 

Answer 2

The function is f(x) = - x² - 9.

The above function is quadratic function, so the domain of the function is { x ∈ R }, where R is real number.

y = - x² - 9

Interchange x and y.

x = - y² - 9

Solve for y.

x  + 9 = - y²

- x  - 9 = y²

y = ± √(- x  - 9).

Replace y by f ^{- 1}(x).

f ^{- 1}(x) = ± √(- x  - 9).

Here f(x) = - x² - 9 (x ≥ 0) considered and the inverse function is f ^{- 1}(x) = + √(- x - 9) (x ≤ 9)

Choose values for x and find the corresponding values for y.

x	y=f(x) = - x2 - 9	(x, y)
0	y = - (0)2 - 9 = -9	(0, -9)
3	y = -(3)2-9 = -9-9=-18	(3, -18)
4	y = -(4)2-9 = -16-9=-25	(4, -25)
5	y = -(5)2-9 = -25-9=-34	(5, -34)

x	y = f - 1(x) = √- x - 9	(x, y)
-9	y = √(-(-9)-9)=√0=0	(-9,0)
-18	y = √(-(-18)-9)=√9=3	(-18,3)
-25	y = √(-(-25)-9)=√16=4	(-25,4)
-34	y = √(-(-34)-9)=√25=5	(-34,5)

 

Answer 3

 

The function f(x) = - x² - 9 is symmetrical about y - axis but any horizontal line crosses the graph at two points, so it is one to one function. Either left or right side will be one to one function about y - axis.

The inverse function f ^{- 1}(x) = + √(- x - 9) is symmetrical about x - axis but domain of the original function is (x ≥ 0) the upper portion will be the invesre function as shown in the figure.

The graph of f ^{- 1}(x) is the reflexion of the graph f (x) in the line y = x. Verify that f ( f ^{- 1}(x) ) = x and f ^{- 1}( f(x) ) = x.

We can further verify this reflective property by testing a few points on each graph.

Note that if the point (a, b) = (0, -9) is on the graph of f and the point (b, a) = (-9, 0) is on the graph of f ^{- 1.}

So, if we consider the function left hand side, then the lower part of the inverse function to be considered (f ^{- 1}(x) = - √(- x - 9) (x ≤ -9))