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Determine whether each function has an inverse function. If it does, find the inverse function and state any restrictions on its domain.

1.
f(x) = |x-6|

2.
f(x) = \sqrt[]{6 -x^(2) }

3.
h(x) = (x+4)/(3x-5)

1 Answer

5 votes
QUESTION 1

The given function is;


f(x) = |x - 6|

This is an absolute value function.

For this function to have an inverse it must be a one-to-one function.

This absolute value function is not one-to-one
because


f( 5)=1
and


f(7) = 1

Since this function has more than one x-value mapping onto one y-value, it is not one-to-one and cannot have an inverse.

You can see from the graph that this function cannot pass the horizontal line test.

QUESTION 2

The given function is


f(x) = \sqrt{6 - {x}^(2) }

Let


y= \sqrt{6 - {x}^(2) }

This implies that,


{y}^(2) + {x}^(2) = 6

We see clearly that, this function is a circle that is centered at the origin.

This means that,


f(x) = \sqrt{6 - {x}^(2) }

is a semicircle.

This function will not pass the horizontal line test and hence does not have an inverse.

QUESTION 3

The given function is


h(x) = (x+4)/(3x-5)

If we put


h(a) = h(b)

We obtain,


(a+4)/(3a-5) = (b+4)/(3b-5)


(a + 4)(3b - 5) = (b + 4)(3a - 5)


3ab - 5a + 12b - 20 = 3ab - 5b + 12a - 20


- 17a = - 17b


a = b

This shows that h(x) has an inverse because it is one-to-one.

Let


y=(x+4)/(3x-5)

We interchange x and y to get,


x=(y+4)/(3y-5)


x(3y - 5) = y + 4


3xy - 5x = y + 4

Group like terms to get,


3xy - y = 5x + 4

Factor to get,


y(3x - 1) = 5x + 4
Solve for y,


y = (5x + 4)/(3x - 1)

Hence


{f}^( - 1)(x) = (5x + 4)/(3x - 1)

where


x \\e (1)/(3)
Determine whether each function has an inverse function. If it does, find the inverse-example-1
Determine whether each function has an inverse function. If it does, find the inverse-example-2
Determine whether each function has an inverse function. If it does, find the inverse-example-3
User Datazang
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