Question

The table below shows some inputs and outputs of the invertible function fff with domain all real numbers. xxx 555 333 111 1 81818 000 999 f(x)f(x)f, left parenthesis, x, right parenthesis 999 -2−2minus, 2 -5−5minus, 5 -1−1minus, 1 111 111111 Find the following values: f^{-1}(f(58))=f −1 (f(58))=f, start superscript, minus, 1, end superscript, left parenthesis, f, left parenthesis, 58, right parenthesis, right parenthesis, equals f(f(5))=f(f(5))=f, left parenthesis, f, left parenthesis, 5, right parenthesis, right parenthesis, equals

Answer 1

Invertible functions are functions whose inverse can be calculated.

The values of the expressions are: $f^{-1}(f(5)) = 5$ and $f(f(5)) = 11$

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:

$f(5) = 9$

Take inverse function of both sides

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

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

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

So, we have:

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

(b) Calculate f(f(5))

In (a) above, as have:

$f(5) = 9$

So, the function becomes

$f(f(5)) = f(9)$

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

So, we have:

$f(f(5)) = 11$

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