HI

A component that provides support for invoking controller action methods directly from JavaScript code using an AJAX request. An <apex:actionFunction> component must be a child of an<apex:form> component.

Used when we need to perform similar action on varioud events. Een though you can use it in place of actionSupport as well where only event is related to only one control.

Example

<!-- Page: -->
	<apex:page controller="exampleCon">
	<apex:form>
		<!-- Define the JavaScript function sayHello-->
		<apex:actionFunction name="sayHello" action="{!sayHello}" rerender="out"
		status="myStatus"/>
	</apex:form>
	
	<apex:outputPanel id="out">
		<apex:outputText value="Hello "/>
		<apex:actionStatus startText="requesting..." id="myStatus">
		<apex:facet name="stop">{!username}</apex:facet>
		</apex:actionStatus>
	</apex:outputPanel>
	
	<!-- Call the sayHello JavaScript function using a script element-->
	<script>window.setTimeout(sayHello,2000)</script>
	<p><apex:outputText value="Clicked? {!state}" id="showstate" /></p>
	<!-- Add the onclick event listener to a panel. When clicked, the panel triggers
	the methodOneInJavascript actionFunction with a param -->
	<apex:outputPanel onclick="methodOneInJavascript('Yes!')" styleClass="btn">
		Click Me
	</apex:outputPanel>
	
	<apex:form>
		<apex:actionFunction action="{!methodOne}" name="methodOneInJavascript"
		rerender="showstate">
			<apex:param name="firstParam" assignTo="{!state}" value="" />
		</apex:actionFunction>
	</apex:form>
</apex:page>

/*** Controller ***/
public class exampleCon {
	String uname;
	public String getUsername() {
		return uname;
	}
	public PageReference sayHello() {
		uname = UserInfo.getName();
		return null;
	}
	public void setState(String n) {
		state = n;
	}
	public String getState() {
		return state;
	}
	public PageReference methodOne() {
		return null;
	}
	private String state = 'no';
}

.ActionSupport : A component that adds AJAX support to another component, allowing the component to be refreshed asynchronously by theserver when a particular event occurs, such as a button click or mouseover.

Used when we want to perform an action on a perticular eventof any control like onchange of any text box or picklist.

Example

<apex:page controller="exampleCon">
	<apex:form>
		<apex:outputpanel id="counter">
			<apex:outputText value="Click Me!: {!count}"/>
				<apex:actionSupport event="onclick"
				action="{!incrementCounter}"
			rerender="counter" status="counterStatus"/>
		</apex:outputpanel>
		
		<apex:actionStatus id="counterStatus"
			startText=" (incrementing...)"
			stopText=" (done)"/>
	</apex:form>
</apex:page>


/*** Controller: ***/
public class exampleCon {
	Integer count = 0;
	public PageReference incrementCounter() {
		count++;
		return null;
	}
	public Integer getCount() {
		return count;
	}
}

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Thanks