The preposition a becomes à (a + article a => à), ao (a + article o), às (a + article => as) and aos (a + article os), see 4.2.1.
If the articles a / as or the demonstrative pronouns aquel / aquelas contracted to à / às / àquele etc. it is called crase. (What is astonishing, because the word is taken from greek and not from latin and means fusion.)
20.1.1 a to mark the indirect object
The indirect object is marked with to in English, if it is placed after the direct object. (He gives John the book. <=> He gives the book to John.)
In Portuguese the indirect object is marked with the preposition a and since there are a lot of indirect object the preposition a meets the article / demonstrative pronoun very often and we get a crase.
O pai dá a maçã à filha.
The father gives the daughter the apple.
In any language we have to clarify the function of a noun. That can be done in different ways and in many languages we have a mix of the different ways possible. (It is always the same sentence: I give the boy an apple.)
1) There is no marking at all. That's how it works in English.
I give the boy the apple. (but: I give the apple to the boy.)
2) The direct object and the indirect object have their own forms. That's how it works in german.
Ich gebe dem Jungen den Apfel.
3) The direct object and the indirect object has a suffix or a prefix. In Persian the direct object is marked by the suffix ra and the indirect object by the suffix be.
من سیب را به این پسر میدهم. Man sib ra be in pesar midaham.
4) The indirect object is marked by a preposition. In roman languages the indirect object is marked, the direct object is not marked at all.
Doy el libro al niño. (Spanish)
5) The indirect object or the direct object are marked by the position in the sentences. If we change the position, we change the meaning.
Thomas looks at Maria. <=> Maria looks at Thomas.
6) Possible in theory, but this author doesn't know any example for that. In theory it is possible to mark the direct / indirect object by the pronunciation.
1) supposes a certain word order and if this word order is not respected, we need a preposition.
2) this is what seems strange to most people of the western hemisphere. The only language these people, people of Nort- / South-America, Western-Europe, come in touch with and that marks the objects by changing the form of the noun is german. Therefore they believe that german is very strange. Actually the opposite is true. Most languages work like german and the roman languages and English are the exceptions.
For 3) there are lot examples, but this author knows only Persian. Quetchua, the language spoken by indigenous people in Bolivia belongs as well to this group.
4) This is the way it works in all roman languages. One can wonder why it is always the indirect object that is marked and never the direct object.
This author doesn't know if there is an example for 5). Perhaps not, because that would be a very inflexible system.
An example for 6) is also unknown to this author and it is not very plausible that they exist, because differences in pronunciation should be made recognisable as well in writing. Furthermore it would be similiar to case 2), because we had different forms, that should be marked with an accent.
In Portuguese the direct object is never marked. A personal accusative like in Spanish, if we put aside a quem, doesn't exist. For the pronouns there are specific forms, the clitic pronouns. Concerning the use of the clitic pronouns see 6.1.1 clitic pronouns.
Verbs can have a direct object and an indirect object, only a direct object or only an indirect object.
direct object / indirect object or only direct object
both are necessary: I give him a book. .
only the direct object is necessary: I sell it to him.
only direct object possible
I see the bird.
We eat the apple.
1) "I give him" as well as "I give it" don't make any sense..
2) I sell it is possible.
In English an indirect object is only possible, if there is a direct object. In English there is therefore no need to know whether a verb requires a direct object or an indirect objects. In roman languages we have to know that. That is not a big thing, because there are only very few verbs that require an indirect object, but it is worth to think about it for five minutes. Lets have a look at these two sentences.
1) the verb requires an indirect object: Informo-lhes que a prova será difícil. ( I informed them that the exam will be difficult.)
2) the verb requires a direct object: Felicitaram-nos por terem chegado bem de viagem. (They congratulated them for having well returned from their trip.)
In both cases we have them in English. Since there is only one object, it is a direct object and even if one would be a direct object and the other one an indirect object, we wouldn't see it, because it is them in both cases. In Portuguese however there is a difference. Informar, to inform, requires an indirect object and felicitar a directe object. We have therefore lhes in the first case and nos in the second case. (nos => em + os. Concerning the adaptations to be made depending on the ending of the verb see 6.1.5.) Therefore we have to know whether the verb requires a direct or an indirect object.
Most verbs in participe require a direct object.
verbs with direct objects
Mariana comeu o bolo.
Marianna eats the cake.
Mas não foi aí que a encontraram.
They didn't find her there.
Demitiu dois empregados por uso indevido do e-mail da companhia.
He fired two employees for use of disallowed use of the internet.
O aluno tem diversos recursos que o auxiliam em seus estudos.
The student has different means supporting him in his studies.
responder a algo / alguém
O aluno respondeu à pergunta da professora.
The student answers the question of the teacher.
Gostaria de me juntar àqueles que já felicitaram o nosso colega.
I want to join the group that has already congratulated our colleagues.
Os países ricos devem ajudar os países pobres.
Rich countries should support poor countries.
Eu disse a ela que eu era fluente em Espanhol.
I told her, said I am fluent in Spanish.
obedecer a alguém
Devemos obedecer aos nossos princípios e ideais.
We have to follow our principles.
esperar a alguém
Miriam esperava a irmã.
Miriam was waiting for her sister.
Ele mentiu a dois empresários.
She lied to two employer.
informar a alguém
Queridos alunos, informo-lhes que a prova será difícil.
Dear students, I inform you that the exam is going to be difficult.