Does a function like this exist? It is decreasing from (-7,6) but it is even everywhere

Yes, such a function can exist. An **even function** is symmetric about the y-axis, meaning \( f(x) = f(-x) \) for all \( x \). A **decreasing function** decreases as \( x \) increases within a specified interval.

Here's how you can construct such a function:

1. **Even and Decreasing in the Given Interval:**
   - To be decreasing from \( x = -7 \) to \( x = 6 \), the function should decrease as \( x \) moves from -7 to 0, and then symmetrically increase from 0 to 6 (due to evenness).
   - For example, consider a function defined by \( f(x) = -|x+1| \) in the interval from \( -7 \) to \( 7 \). This function is even and decreases as \( x \) increases from -7 to 0, then increases from 0 to 7.
   
2. **Constructing an Example:**
   - Let \( f(x) = -(x^2 + c) \) for some constant \( c \). This function is even (because \( f(x) = f(-x) \)), and it decreases for negative \( x \) values and increases for positive \( x \) values, meeting the requirement of being decreasing in the interval \( (-7, 6) \).

3. **Adjusting the Function:**
   - To fit specific conditions such as decreasing from \( x = -7 \) to \( x = 6 \) exactly, you might have to adjust the constant or use a piecewise function to define different behaviors in different intervals.

Would you like more details on constructing this function, or do you have any other questions?  

Here are some related questions you might find interesting:

1. What is the difference between even and odd functions?
2. Can a function be both increasing and even?
3. How can we determine if a function is even or odd?
4. What does it mean for a function to be strictly decreasing?
5. Can you give an example of a function that is decreasing but not even?

**Tip:** When dealing with even functions, remember that their graph is always symmetric about the y-axis.

Learn how to construct a function that is both even and decreasing over the interval (-7, 6). This involves understanding the symmetry of even functions and the behavior of decreasing functions within specified intervals. Examples and methods for constructing such functions are explored.

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