The table below shows some inputs and outputs of the invertible function f with domain all real numbers.

Part 1

-9

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Explanation:

Let's focus on evaluating   first. We need to look at the table and locate f(x) = 13. So we're looking in the f(x) row where 13 shows up. That's in the second to last column. Right above that we have x = 5

This means f(5) = 13 and   The inverse undoes the f(x) function. So the input x and output y values swap places. This is why we're reading the table in reverse.

We can replace all of  with 5 to go from to

Then we repeat the process of using the table. Locate 5 in the f(x) row. This is in the last column. The value above that is x = -9.

So f(-9) = 5 and

Overall,

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Part 2

 -13

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Explanation:

Same idea as before. Locate 8 in the f(x) row. The value above this is x = -13

This means f(-13) = 8 and

f⁻¹(-2) = 3 and f⁻¹(1) = 0.

Step-by-step explanation:

By the definition of inverse functions:

From the table, note that f(3) = -2.

Then by definition, f⁻¹(-2) = 3.

Likewise, f(0) = 1.

Then by definition, f⁻¹(1) = 0.

Invertible functions are functions whose inverse can be calculated.

The values of the expressions are: and

The table entry is given as:

x | 5 | 3 | 1 | 18 | 0 | 9

f(x) | 9 | -2 | -5 | -1 | 1 | 11

(a) Calculate f^-1(f(5))

Start by calculating f(5).

From the table, we have:

Take inverse function of both sides

From the table, the inverse value of 9 is 5.

i.e.

So, we have:

(b) Calculate f(f(5))

In (a) above, as have:

So, the function becomes

From the table, the value of f(9) is 11.

So, we have:

Read more about invertible functions at:

link

Step-by-step explanation:

The given table :

x: -14,-7,-12,9,10,-2

f(x):11,-12,5,1,-2,13

Since f is invertible ( given) , then  exists.

Now , from table  [ x= 9 corresponding to f(x) =1]

                           [ f(x) = 11 corresponding to x=-14]

then,

So,

Also, x= 10 corresponding to f(x) =-2, then

( 1 ) ,

( 2 )

Step-by-step explanation:

Taking the function and it's inverse, it should be the following property -

- this therefore makes the functionality " " equivalent to 576 itself.

For this second part here let's consider the portion " " firstly. As you can see from the table, - 7 should represent the x - value, hence . Now for this second bit here, we have to take the inverse - making the what should have been the x - value, now the f( x ) value. Therefore, .

This would make = = 13.

Step-by-step explanation:

Given

The attached table

This represents an inverse function.

So, we look into x row for its value.

i.e.

Just like (a)

---- by looking into the x rows

So:

Answer for the first box = 10

Answer for the second box =  -10

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Work Shown:

f(x) is a function where we plug in the given row of x values to have them lead to the corresponding f(x) values in the table. We see that x = 14 leads to y = f(x) = 7. Going backwards, we can say . The inverse simply undoes everything f(x) does.

So,

For the portion , look at the column that has x = 10. This shows it leading to f(x) = 14. You trace this column upward to go backward to undo f(x).

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That takes care of the first part. The second part is pretty much the same idea. We locate where -4 is in the bottom row, and then we write down the corresponding x value.

We see that y = -4 pairs with x = -10, which is why

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Keep in mind that this function must be one-to-one for the inverse to exist. If not, then you'll have to do some kind of domain restriction.

The value of is 3

The value of is 18

Explanation:

Given that the set of inputs and outputs of the invertible function f.

We need to determine the values of and

The value of :

We need to determine the image of the y - value -2.

The image of the y - value (-2) is the corresponding x - value from the given table.

Thus, from the table, the image of -2 is 3.

Hence, the value of is 3.

The value of :

We need to determine the image of the y - value -1.

The image of the y - value (-1) is the corresponding x - value from the given table.

Thus, from the table, the image of -1 is 18

Hence, the value of is 18.

f⁻¹(-2) = 3 and f⁻¹(1) = 0.

Step-by-step explanation:

By the definition of inverse functions:

From the table, note that f(3) = -2.

Then by definition, f⁻¹(-2) = 3.

Likewise, f(0) = 1.

Then by definition, f⁻¹(1) = 0.

Step-by-step explanation:

Given

The attached table

This represents an inverse function.

So, we look into x row for its value.

i.e.

Just like (a)

---- by looking into the x rows

So: