# Find the average value of a function

• ~Sam~
In summary, to find the average value of the function f(x)= |x+1| sgnx on the interval [-2,2], you would split the function into three intervals: [-2,-1], [-1,0], and [0,2]. Then, integrate each interval using the given function and combine the results to find the average value.
~Sam~

## Homework Statement

Find the average value of a function of f(x)= |x+1| sgnx on the interval [-2,2]

## Homework Equations

The average value formula.

## The Attempt at a Solution

I know I can divide |x+1| into two integrals from [-2,0] and [0,2] (I think?? can anyone confirm this is the proper split?) and add and solve. But what can I do about sgnx?

Last edited:
I would split |x+1| into two intervals. x<(-1) and x>=(-1). And sgn(x) I would split at x=0. You might want to think about splitting |x+1|*sgn(x) into three intervals [-2,-1], [-1,0] and [0,2].

Dick said:
I would split |x+1| into two intervals. x<(-1) and x>=(-1). And sgn(x) I would split at x=0. You might want to think about splitting |x+1|*sgn(x) into three intervals [-2,-1], [-1,0] and [0,2].

Ohh I see why you would split into three pieces. But...if I do split sgn(x) at x=0, which would mean [-2,0] and [0,2] and evaluate with |x| as my integral, wouldn't I get 0? (I know I'm suppose to get 4)

No, [-2,0] and [0,2} is not three pieces! Dick said to use [-2,-1], [-1,0], and [0,2].
On [-2,-1], x+1< 0 so |x+1|= -(x+1). x< 0 so sgn(x)= -1. |x+1|sgn(x)= -(x+1)(-1)= x+1. Integrate that from -2 to -1.

On [-1,0], x+1> 0 so |x+1|= x+1. x< 0 so sgn(x)= -1. |x+1|sgn(x)= (x+1)(-1)= -(x+1). Integrate that from -1 to 0.

On [0,1], x+1> 0 so |x+1|= x+1. x>0 so sgn(x)= 1. |x+1|sgn(x)= (x+1)(1)= x+ 1. Integrate that from 0 to 2.

## 1. What does it mean to find the average value of a function?

Finding the average value of a function refers to calculating the value that represents the average or mean of all the values that the function takes on within a given interval.

## 2. How do you find the average value of a function?

To find the average value of a function, you need to integrate the function over the given interval and then divide the result by the length of the interval.

## 3. What is the formula for finding the average value of a function?

The formula for finding the average value of a function is: (1/b-a) ∫f(x)dx, where a and b represent the lower and upper limits of the interval, and f(x) is the given function.

## 4. Why is finding the average value of a function important?

Calculating the average value of a function can provide useful information about the overall behavior of the function within a given interval. It can also be used to approximate the value of a function at a specific point.

## 5. Can the average value of a function be negative?

Yes, the average value of a function can be negative, as it is simply the average or mean of all the values that the function takes on within a given interval. However, if the function has negative values, the average value may also be negative.

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