Chapter 1.6, Problem 115E

a.

To determine

To state : whether the function is one-to-one.

Answer to Problem 115E

Yes, the function is one-to-one.

Explanation of Solution

Given information :

Thetable provided in the question,

Men’s U.S. shoe size	Men’s European shoe size
8	41
9	42
10	43
11	44
12	45
13	46

from the table it can be observed that for each value of $x$ there is unique value of $y$.

Thus, the function is one-to-one.

b.

To determine

To find : the value of $f\left(11\right)$.

Answer to Problem 115E

The value of $f\left(11\right)$ is $44$.

Explanation of Solution

Given information :

The table provided in the question,

Men’s U.S. shoe size	Men’s European shoe size
8	41
9	42
10	43
11	44
12	45
13	46

Calculation : from the table it can be observed that when $x=11$ , the value of $f\left(x\right)=44$ ,

Thus, the value of $f\left(11\right)$ is $44$.

c.

To determine

To find : the value of $f^{-1}\left(43\right)$.

Answer to Problem 115E

The value of $f^{-1}\left(43\right)$ is $10$.

Explanation of Solution

Given information :

The table provided in the question,

Men’s U.S. shoe size	Men’s European shoe size
8	41
9	42
10	43
11	44
12	45
13	46

Calculation : from the table it can be observed that when $x=10$ , the value of $f\left(x\right)=43$ ,

So for inverse of function when $f^{-1}\left(43\right)$ then the value of $10$.

Thus, the value of $f^{-1}\left(43\right)$ is $10$.

d.

To determine

To find : the value of $f\left(f^{-1}\left(41\right)\right)$.

Answer to Problem 115E

The value of $f\left(f^{-1}\left(41\right)\right)$ is $41$.

Explanation of Solution

Given information :

The table provided in the question,

Men’s U.S. shoe size	Men’s European shoe size
8	41
9	42
10	43
11	44
12	45
13	46

Calculation : from the table it can be observed that when $x=8$ , the value of $f\left(x\right)=41$ ,

So for inverse of function when $f^{-1}\left(41\right)$ then the value of $8$.

  $\begin{array}{l}f\left(f^{-1}\left(41\right)\right)=f\left(8\right)\text{ [}{f}^{-1}\left(41\right)=8]\\ f\left(f^{-1}\left(41\right)\right)=41\text{ [}f\left(8\right)=41]\end{array}$

Thus, the value of $f\left(f^{-1}\left(41\right)\right)$ is $41$.

e.

To determine

To find : the value of $f^{-1}\left(f\left(12\right)\right)$.

Answer to Problem 115E

The value of $f^{-1}\left(f\left(12\right)\right)$ is $12$.

Explanation of Solution

Given information :

The table provided in the question,

Men’s U.S. shoe size	Men’s European shoe size
8	41
9	42
10	43
11	44
12	45
13	46

Calculation : from the table it can be observed that when $x=12$ , the value of $f\left(x\right)=45$ ,

  $\begin{array}{l}{f}^{-1}\left(f\left(12\right)\right)=f^{-1}\left(45\right)\text{ [}f\left(12\right)=45]\\ f^{-1}\left(f\left(12\right)\right)=12\text{ [}{f}^{-1}\left(45\right)=12]\end{array}$

Thus, the value of $f^{-1}\left(f\left(12\right)\right)$ is $12$.