Question

Evaluate inverse functions
The table below shows some inputs and outputs of the invertible function f with domain all real numbers.
x. |-7|11|-13|6|5|-9|
f(x). 7|12|8|-7|13|5|
Find the following values:
f-1(F-1(13)) =
f-1(8) =

Answer 1

Part 1

Answer: -9

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Explanation:

Let's focus on evaluating $f^{-1}(13)$ 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 $f^{-1}(13) = 5$. 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 $f^{-1}(13)$ with 5 to go from $f^{-1}(f^{-1}(13))$ to $f^{-1}(5)$

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 $f^{-1}(5) = -9$

Overall, $f^{-1}(f^{-1}(13)) = -9$

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Part 2

Answer:  -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 $f^{-1}(8) = -13$