For example, the noun 'truth' has an adjective 'true' which may be illustrated with "a statement that contains the truth is a true statement".
Another example could be the noun 'day' and its adjective 'daily' which may be illustrated with "each day we wash the dinner dishes, so washing the dishes is a daily chore".
The English words 'age' and 'eon' are nouns that refer to a period of time, often a very long period of time, which has a beginning and an end.
Unfortunately in English we don't have a simple single word adjective corresponding to the noun 'age', although 'eon' does have the rarely used 'eonian'.
The closest we can come to a "single word" adjective for the noun 'age' might be 'age-during' or 'age-lasting', or we might try to use an adjectival phrase like 'belonging to the age', 'pertaining to the age', 'during the age', 'of the age' or 'for the age'.
As difficult as it is to find an adjective for age that we are comfortable with, we should never use words like 'forever' or 'everlasting' - words that convey the idea of never-ending, nor a word like 'eternal' - a word that conveys the idea of no beginning and no end.
In the New Testament, the Greek noun that translates into age or eon in English is 'aion' and its corresponding adjective is 'aionian'. Possible uses might be: We are living in the aion of the church, so we are the aionian church, and, Believers will live and reign with Christ in the coming aion, so they will have aionian life.
Unfortunately, most of our popular English Bibles quite wrongly translate 'aionian' in the Greek as 'eternal' in English. And this error causes us to miss the distinction between eonian life (life during the ages) and eternal life (life after the ages have concluded).
Here are a few examples of using eonian correctly.
Just as Moses lifted up the snake in the desert, so the Son of Man must be lifted up, that everyone who believes in him should not perish but have eonian life. For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eonian life.
[John 3 : 14 - 16 NIV corrected]
Very truly I tell you, whoever hears my words and believes him who sent me has eonian life and will not be judged but has crossed over from death to life. [John 5 : 24 NIV corrected]
“They also will answer, ‘Lord, when did we see you hungry or thirsty or a stranger or needing clothes or sick or in prison, and did not help you?’
“He will reply, ‘Truly I tell you, whatever you did not do for one of the least of these, you did not do for me.’“Then they will go away to eonian punishment, but the righteous to eonian life.”
[Matthew 25 : 44 - 46 NIV corrected]
As you know, because of Jesus' death on the cross, everyone will receive eternal life eventually (at the end of the ages), but only believers in the gospel will receive eonian life in the coming millennium, illustrating that although God is the Saviour of the world, there is something special for believers. [see 1 Timothy 4 : 10]
That special something is eonian life, life that continues through the coming eons.
So correct grammar reveals the difference between eonian life and eternal life - between life in the kingdom ages for believers and life in eternity beyond the ages for everyone. Indeed, without this correction, it is difficult to find the millennial kingdom anywhere other than in Revelation 20.
(I am beginning to use 'eonian' more and more in recent times, instead of the other age-related options, because it is just one word and because it closely resembles the Greek 'aionian'. Having said that, I am tempted to create my own word: age-ian, which looks very strange but is growing on me.)