There are 4 odd functions and 3 even functions in the gallery of the 12 basic functions. After multiplying these functions together pairwise in different combinations and exploring their graphs, make a conjecture about the summetry of: A) a product of 2 odd functions ( in other words is their product even, odd, or neither) B) a product of 2 even functions C) a product of an odd and even function

Remember the definitions:

even: f(-x) = f(x)
odd: f(-x) = -f(x)

check your answers with some simple monomials, such as x^2 and x^3

To answer these questions, let's analyze the properties of odd and even functions.

An odd function is defined as f(x) = -f(-x), which means that its graph is symmetric about the origin (0,0). Examples of odd functions include f(x) = x, f(x) = x^3, f(x) = sin(x), etc.

On the other hand, an even function is defined as f(x) = f(-x), which means that its graph is symmetric about the y-axis. Examples of even functions include f(x) = x^2, f(x) = |x|, f(x) = cos(x), etc.

Now, let's explore the combinations of these functions:

A) Product of 2 odd functions:
If we multiply two odd functions together, say g(x) and h(x), we get f(x) = g(x) * h(x). Since both g(x) and h(x) are odd functions, we can substitute -x for x and observe the symmetry:

f(-x) = g(-x) * h(-x)
= -g(x) * -h(x)
= g(x) * h(x)
= f(x)

From this observation, we can conclude that the product of two odd functions is an even function.

B) Product of 2 even functions:
If we multiply two even functions together, say g(x) and h(x), we get f(x) = g(x) * h(x). Since both g(x) and h(x) are even functions, we can substitute -x for x and observe the symmetry:

f(-x) = g(-x) * h(-x)
= g(x) * h(x)
= f(x)

From this observation, we can conclude that the product of two even functions is an even function.

C) Product of an odd function and an even function:
If we multiply an odd function, say g(x), and an even function, say h(x), together, we get f(x) = g(x) * h(x). Since g(x) is odd and h(x) is even, we can substitute -x for x and observe the symmetry:

f(-x) = g(-x) * h(-x)
= -g(x) * h(x)
= -f(x)

From this observation, we can conclude that the product of an odd function and an even function is an odd function.

In summary:

A) The product of two odd functions is an even function.
B) The product of two even functions is an even function.
C) The product of an odd function and an even function is an odd function.

To make conjectures about the symmetry of products of odd and even functions, we first need to understand the properties of odd and even functions.

An odd function is defined as a function that satisfies the property f(-x) = -f(x). In other words, if you replace x with its opposite value, the function will have the opposite sign. Examples of odd functions include f(x) = x, f(x) = x^3, f(x) = sin(x).

An even function, on the other hand, is a function that satisfies the property f(-x) = f(x). This means that if you replace x with its opposite value, the function remains the same. Examples of even functions include f(x) = x^2, f(x) = |x|, f(x) = cos(x).

Now let's explore the symmetry of product combinations:

A) Product of two odd functions:
Let's consider two odd functions, f(x) and g(x). When we multiply them together, we have (f * g)(x) = f(x) * g(x). To determine the symmetry of this product, we substitute -x into the equation:
(f * g)(-x) = f(-x) * g(-x) = -f(x) * -g(x) = f(x) * g(x).
Since the product of two negative quantities equals the product of the positive counterparts, we can see that the product of two odd functions is an even function.

B) Product of two even functions:
Similarly, let's consider two even functions, f(x) and g(x). Their product is given by (f * g)(x) = f(x) * g(x). When we substitute -x into the equation:
(f * g)(-x) = f(-x) * g(-x) = f(x) * g(x).
In this case, since even functions remain the same when x is replaced with -x, we can conclude that the product of two even functions is also an even function.

C) Product of an odd and an even function:
Lastly, let's examine the product of an odd function, f(x), and an even function, g(x). Their product is given by (f * g)(x) = f(x) * g(x). When we substitute -x into the equation:
(f * g)(-x) = f(-x) * g(-x) = -f(x) * g(x).
As we can observe, when multiplying an odd function by an even function, the resulting product is an odd function. This is due to the negative sign that arises from multiplying the two functions together.

In summary:
A) A product of two odd functions is an even function.
B) A product of two even functions is an even function.
C) A product of an odd and an even function is an odd function.