"Exhort slaves to be putting themselves in subjection to their own masters with implicit obedience" (Wuest)

Urge is not present in the Greek but is added by the NASB translators. The text reads literally

Bondslaves (1401) (doulos) refers to one born into slavery in contrast to a person taken in war and sold as a slave (andrapodon). Slaves formed a large class of society in Paul's day.

Doulos conveys the idea of the slave's close, binding ties with his master, belonging to him, obligated to do his will and in a permanent relation of servitude to him. (Click or here for more  discussion of doulos).

To be subject (5293) (hupotasso from hupó = under + tasso = arrange in orderly manner, assign or dispose to a certain position or to a particular task) (Click word study on hupotasso) means to cause to be in a submissive relationship, to subordinate or bring one under the firm control of another. Hupotasso was a military term describing soldiers lining up under or rank under their authority. Hupotasso also referred to the arrangement of military implements on a battlefield in such a way as to facilitate effective warfare. 

Hupotasso focuses on function, not essence. Submission to political authorities (see note Romans 13:1), wives to husbands (see note  Colossians 3:18), young to elders (see note 1 Peter 5:5).

In non-military use, hupotasso refers to a voluntary attitude of placing one's self under the authority of another.  In the present context, the bondslave is exhorted to continually (present tense) place himself or herself (middle voice = reflexive or if passive to be subjected to) in subjection to the master who for the most part were pagans. Submission in such potentially adverse circumstances represents an act of faith. The born again bondslave would be trusting God to direct his or her life and to work out His purposes in His time. Clearly, there is a danger in submitting to others, especially if they have a different world view (pagan or Cretan in this case) for they might take advantage of us. Ultimately however their submission to their master, pagan or not, would reflect submission to God's authority and He is responsible for the care.

How is it possible to surrender one's rights to a master in authority?

In Ephesians Paul writes that believers should "not get drunk with wine, for that is dissipation, but be (continually) filled with (controlled by) the Spirit" (Eph 5:18) and then should "be subject to one another in the fear of Christ" (Eph 5:21) Slaves filled with the Spirit would be enabled to obey the spirit of the law to the glory of God.

Masters (1203) (despotes) (Click word study of despotes) meaning master, lord, or one who has absolute, undisputed ownership of and uncontrolled power over another. Five of the nine uses in the NT refer to the master of the house who had unrestricted authority.

IN EVERYTHING TO BE WELL-PLEASING: en pasin euarestous einai (PAN):

"in all things; to give them satisfaction" (Wuest)

To be (1511) (eimi = to be, present tense = continuous action, active voice = subject makes volitional choice to carry out the action)

Everything (3956) (pas) means no exceptions. Notice it precedes the verb emphasizing the importance of "no exceptions" It means whether they treat you well or not. God's children are never to return evil for evil or insult for insult to anyone, so that they may show a proper opinion of their Father and the supernatural life He gives us to the lost world.

Why is this so critical that bondslaves (and all spiritual bondslaves of Christ) adorn (make attractive) the doctrine of God our Savior? In an article in Breakpoint entitled "Drawn to the Light - Why Muslims Convert to Christianity" read and be convicted/motivated...

Dr. Dudley Woodberry, professor of Islamic Studies... aware that throughout the world Muslims have been turning to Christ, was curious about the reasons why -- especially in countries where the cost of converting is so high. To find the answer, he created a detailed questionnaire. Over a 16-year period, some 750 Muslims from 30 countries filled it out -- and the results are eye-opening. The number one reason Muslim converts listed for their decision to follow Christ was the lifestyle of the Christians among them.

As Woodberry, Russell Shubin, and G. Marks write in CHRISTIANITY TODAY, Muslim converts noted that "there was no gap between the moral profession and the practice of Christians" they knew. An Egyptian convert contrasted the love shown by Christians "with the unloving treatment of Muslim students and faculty he encountered at a university in Medina." Other converts were impressed that "Christians treat women as equals" and enjoy loving marriages. And poor Muslims observed that "the expatriate Christian workers they knew had adopted, contrary to their expectations, a simple lifestyle." They wore locally made clothes and abstained from pork and alcohol, so as not to offend Muslim neighbors. (Read the full discussion - Drawn to the Light - Breakpoint)

Well pleasing (2101) (euarestos from eu = good, well + arestos = pleasing, desirable, proper, fit, agreeable from arésko = to please or be pleasing/acceptable to)  means that which causes someone to be pleased. It is something which is well approved, eminently satisfactory, or extra-ordinarily pleasing.

Euarestos is used 9 times in the NT in the NASB (3x Ro; 1x 2Cor; 1x Eph; 1x Phil; 1x Col; 1x Titus; 1x Heb) and is translated as: acceptable, 3; pleasing, 3; well-pleasing, 3. Note that the predominant usage refers to well-pleasing to God and most often describes the goal of a believer's life.

Romans 12:1 (note) I urge you therefore, brethren, by the mercies of God, to present your bodies a living and holy sacrifice, acceptable to God, which is your spiritual service of worship.

Romans 12:2 (note) And do not be conformed to this world, but be transformed by the renewing of your mind, that you may prove what the will of God is, that which is good and acceptable and perfect.

Romans 14:18 (note) For he who in this way serves Christ is acceptable to God and approved by men.

2 Corinthians 5:9 Therefore also we have as our ambition, whether at home or absent, to be pleasing to Him.

Ephesians 5:10 (note) trying to learn what is pleasing to the Lord.

Philippians 4:18 (note) But I have received everything in full, and have an abundance; I am amply supplied, having received from Epaphroditus what you have sent, a fragrant aroma, an acceptable sacrifice, well-pleasing to God.

Colossians 3:20 (note) Children, be obedient to your parents in all things, for this is well-pleasing to the Lord.

Titus 2:9 (note) Urge bondslaves to be subject to their own masters in everything, to be well-pleasing, not argumentative,

Hebrews 13:21 (note) equip you in every good thing to do His will, working in us that which is pleasing in His sight, through Jesus Christ, to whom be the glory forever and ever. Amen.

These bondslaves are to be continually well approved, eminently satisfactory, or extra-ordinarily pleasing to their masters. 

It is possible to obey externally, but not “from the heart” (Eph 6:6). It is possible to do a job grudgingly. This exhortation however would not be easy to comply with as some unsaved masters would not be thoughtful and would overwork their slaves.

In Romans we can see a sense of what euarestos means where Paul writes that the will of God is

good and acceptable (euarestos) and perfect.  (see note Romans 12:2)

God's will is well-pleasing because you cannot add anything to the will of God and in any way improve it. You could not take anything away from it and make it better. God's will is totally acceptable. And this is the attitude and actions Paul is calling for in servants here in Titus and in every saint in Romans 12 where he exhorts us

"by the mercies of God, to present your bodies a living and holy sacrifice, acceptable (euarestos) to God, which is your spiritual service of worship." (see note Romans 12:1)

In using euarestos Paul borrows from Old Testament sacrificial language to describe the kind of holy living that God approves, a “living sacrifice” that is morally and spiritually spotless and without blemish, and which He finds well-pleasing.

In his second letter to the Corinthians Paul writes that

"we have as our ambition (Greek root means loving what is honorable), whether at home or absent, to be pleasing (euarestos) to Him." (2Cor 5:9).

Paul's service on earth was designed to bring pleasure to the heart of his Lord, whether Paul was still here on earth or whether he was standing before the Judgment Seat of Christ. To be well pleasing was Paul’s highest goal, and should also be for every believer.

Writing to the saints at Ephesus Paul encouraged them to be "trying to learn (idea of testing or proving to learn by clear, convincing evidence what is truly honoring to God)  what is pleasing to the Lord." (Eph 5:10) In other words, these saints were to be putting every thought, word, and action to the test to discern "What does the Lord think about this?" "How does this appear in His presence?" Every area of our life should come under this searchlight, our...conversation, standard of living, clothes, books, business, pleasures, web surfing habits, friendships, sports, etc. The ultimate question should be... Will it be well pleasing (euarestos) to the Lord?

NOT ARGUMENTATIVE: me antilegontas (PAPMPA):

not answering again (KJV)

not talk back (NLT)

They are not to "answer back" (Phillips)

not crossing them (Wuest)

Argumentative (483) (antilego from anti = against + lego = speak)  literally not speaking against. Modern slang might say "not talking back", contradicting, disputing, refusing, disobeying. The present tense indicates this a habitual practice.

not purloining (KJV)

they are not...to be light fingered (Phillips)

Not taking what is not theirs (BBE)

Not defrauding (DRA)

and there must be no pilfering (NJB)

Nor to steal by taking things of small value (Amp)

Pilfering (3557) (nosphizo from nósphi = apart, separated) conveys the idea is stealing in small quantities or practicing petty theft. KJV translates it as purloin (from Middle English, to put away, misappropriate) which means to appropriate wrongfully and often by a breach of trust. Whereas pilfer implies stealing repeatedly in small amounts, whereas purloin stresses removing or carrying off for one’s own use or purposes.

Nosphizo is used 3 times in the NT in the NASB (Acts 2x; Titus) and is translated as keep back, 1; kept back, 1; pilfering (KJV = purloin), 1.

Nosphizo means to keep back or misappropriate something for one's self that which should or does belong to someone else. This term was found in secular Greek writings in reference to embezzlement of public treasures. It speaks of secretly appropriating and setting apart for one’s self the property of another.

Nosphizo is used twice in Acts 5 when Peter confronted Ananias who had

kept back (nosphizo - Amplified adds "and wrongfully appropriated") some of the price for himself, with his wife's full knowledge, and bringing a portion of it, he laid it at the apostles' feet. But Peter said, "Ananias, why has Satan filled your heart to lie to the Holy Spirit and to keep back (nosphizo) some of the price of the land?" (Acts 5:2-3)

A notorious OT use of nosphizo is found in the Greek translation of the Hebrew (Lxx = Septuagint) passages in Joshua 7 where we read  that

"the sons of Israel acted unfaithfully in regard to the things under the ban (a vow or pledge under which property or persons devoted to pagan worship were destroyed), for Achan, the son of Carmi, the son of Zabdi, the son of Zerah, from the tribe of Judah, took some (Greek = nosphizo, Heb = laqah = grasped or seized) of the things (the spoils of battle including here a mantle, silver and gold) under the ban, therefore the anger of the Lord burned against the sons of Israel.” (Joshua 7:1)

From observing what happened to Ananias and his wife in Acts and to Achan and his family in Joshua, pilfering from the Lord is neither prudent or profitable!

Vincent has the following note regarding the meaning of nosphizo...

Purloining (nosphizomenous). Only here and Acts 5:2-3. LXX (Septuagint), Josh. 7:1; 2 Macc. 4:32. Often in Classic Greek. From nosphi apart. The fundamental idea of the word is to put far away from another; to set apart for one’s self; hence to purloin and appropriate to one’s own use. Purloin is akin to prolong: prolongyn or purlongyn “to put fer awey.” Old French, porloignier or purloignier." (Vincent, M. R.  Word Studies in the New Testament Vol. 4, Page 343)

Another source says nosphizo means to "separate surreptitiously or clandestinely for one's own use".

The verb is present tense (continuous action) and in the middle voice which is reflexive ("yourself").

With the negative Paul is saying don't (continually) keep setting apart for yourself (reflexive) a part of that which has been entrusted to your care by your master. Don't embezzle or appropriate wrongfully some of the Master's good for your own use as did

"a man named Ananias, with his wife Sapphira" who after they had "sold a piece of property...kept back some of  (nosphizo) the price for himself, with his wife's full knowledge (Acts 5:2,3 5:3 for the consequences of their actions read Acts 5).

Vincent says that

The fundamental idea of the word is to put far away from another; to set apart for one's self; hence to purloin (to take or carry away for one's self) and appropriate to one's own use.

BUT SHOWING ALL GOOD FAITH: alla pistin pasan endeiknumenous (PMPMPA) agathen: (Ge 31:37;31:38  39:8 39:9; 1Sa 22:14; 26:23; Ps 101:6; Mt 24:45; Lu 16:10; 1Cor 4:2)

but to show that they can be fully trusted (NIV)

but they must show themselves to be entirely trustworthy and good (NLT)

And they should show their masters that they can be fully trusted (ICB)

Instead, tell slaves to show their masters how good and completely loyal they can be" (GWT)

but giving clear signs of their good faith (BBE)

but must show themselves to be entirely trustworthy (TLB)

but to prove themselves truly loyal and entirely reliable and faithful throughout (Amp)

they must show complete honesty at all times (NJB)

Showing (1731) (endeíknumi from en = in, to + deíknumi = explain meaning or significance of something by demonstration) means to indicate by word or act. The present tense indicates this should be their habitual practice.

Regenerate, redeemed slaves are to be continually (middle voice) demonstrating, showing forth or manifesting words and works that show they are loyal and can be fully trusted. They were continually proving themselves to be faithful servants in everything they are entrusted with.  In a parallel teaching to the Corinthians, Paul wrote that

"it is required of stewards that one be found trustworthy." (1 Cor 4:2)

How might this spiritual dynamic work in everyday practice? Masters know that the slaves could and would steal and God says our sin will find us out so the masters surely knew who to trust with cleaning the expensive silverware and dusting the bottles of fine wine. The point is that when the master saw such integrity and fidelity in a slave, he would have been intrigued.  What made this lowly man or woman tick different than the average slave who thought little of cutting off for himself a small portion of the fine rib eye steak the master & mistress were to dine on that night! Ultimately these pagan masters would see through the "beautiful lives" of these slaves, the beauty of the teaching about the the One to Whom the slave belonged. Let us conduct ourselves in a manner worthy of the Gospel so that our lives would be like attractive ornaments that draw attention to the "Tree" of Calvary's love!

Matthew Henry writes that

"Faithfulness in a servant lies in the ready, punctual, and thorough execution of his master's orders; keeping his secrets and counsels, dispatching his affairs, and managing with frugality, and to as much just advantage for his master as he is able; looking well to his trusts, and preventing, as far as he can, all spoil, or loss, or damage."

THAT THEY MAY ADORN THE DOCTRINE OF GOD OUR SAVIOR IN EVERY RESPECT: hina ten didaskalian tou soteros humon theou kosmosin (3PPAS) en pasin : (Mt 5:16; Eph 4:1; Php 1:27; Phil 2:15, 16; 4:8; 1Pe 2:12; 3:16) (1Ti 5:17; 6:1 6:3; 2 Jn 1:9) 

so that in every way they will make the teaching about God our Savior attractive" (NIV)

Then they will make the teaching about God our Savior attractive in every way" (NLT)

so that in everything they may be an ornament to the doctrine of God our Savior" (NRSV)

Then, in everything they do they will make the teaching of God our Savior attractive" (ICB)

Then they will show the beauty of the teachings about God our Savior in everything they do" (GWT)

in all things doing credit to the teaching of God our Saviour" (BBE)

In this way they will make people want to believe in our Savior and God" (TLB)

so that in everything they may be an ornament and do credit to the teaching [which is] from and about God our Savior" (Amp)

That (2443) (hina) is a purpose clause explaining the reason for or the purpose they are to behave as he has prescribed. That they might, in their lives, “beautify the Bible,” making it attractive to unbelievers.

Paul had similar instructions for Timothy writing

"Let all who are under the yoke as slaves regard their own masters as worthy of all honor so that (purpose clause) the name of God and our doctrine may not be spoken against.." (1Ti 6:1)

Adorn (2885) (kosmeo from kósmos = adorning or order, ornament, decoration, adornment -- this root word gives us our English "cosmetic" something most women use to "adorn" their face and make themselves more physically attractive) speaks of that which is congruous, fitting, orderly and in keeping with the simplicity, purity and beauty of our Lord Jesus, radiating from a new heart in Christ and a transformed mind of Christ. Kosmeo in this verse then refers to the behavior of bond slaves which was to be such that it caused the truth about God to be beautifully "decorated".  Kosmeo conveys the idea of arranging something in proper order so as to give it symmetry, comeliness, and beauty. It was now in order and appeared neat, well organized and tidy and so was attractive.

The present tense indicates this a habitual practice, continually showing the beauty of Christ in them the hope of glory, for the world is always looking for a flaw in the professed Christian, be they an employee or employer.

Kosmeo is used 10 times in NT in the NASB (3x Mt; 2x Lu; 1x 1Ti; 1x Titus; 1x Pe;