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

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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.

asked May 26, 2014 in PRECALCULUS by bilqis Pupil

3 Answers

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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 :

graph the f(x)=-x^2-9

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.

 

answered May 27, 2014 by steve Scholar
+1 vote

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)

 

answered May 31, 2014 by joly Scholar
+1 vote

 

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))

 

answered May 31, 2014 by joly Scholar

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