Suppose f(x) is an odd function and g(x) is an even...

Suppose $f(x)$ is an odd function and $g(x)$ is an even function. Fill in the missing cntrics in the table.
$$\begin{array}{c|c|c|c|c|c}
x & -2 & -1 & 0 & 1 & 2 \\
\hline f(x) & & & 0 & -2 & \\
\hline g(x) & 0 & 2 & 1 & & \\
\hline(f \circ g)(x) & & 1 & -2 & & \\
\hline
\end{array}$$

Key Concepts

Odd Function

An odd function is one that satisfies the condition f(-x) = -f(x) for every x in its domain. This means the function has rotational symmetry about the origin, and the value of the function for any negative input is the opposite of the value for the corresponding positive input.

Even Function

An even function is characterized by the property g(-x) = g(x) for all x in its domain, implying that the graph of the function is symmetric about the y-axis. This symmetry allows one to determine values for negative inputs directly from the corresponding positive inputs.

Function Composition

Function composition, denoted as (f ? g)(x) = f(g(x)), involves applying one function to the result of another. Understanding the individual properties of the functions, such as being odd or even, can be crucial in determining the outcome of the composition, as the symmetries of the original functions can influence the composition.

Transcript

00:02 In this problem, we're looking at composition of functions, and we're using this table of values to see if we can fill in the rest of the values that are missing in this table.

00:10 So the problem told us that f of x is an odd function and that g of x is an even function.

00:16 So first of all, for f of x, if it's an odd function, that means that it's symmetrical, but that the the numbers on each side, one will be positive and one will be negative.

00:26 So if f of positive one is negative two, then f of negative one is going to be a positive two.

00:33 And if we knew f of two, we could also find f of negative two, but we don't know that.

00:37 So let's see if we can find some of these missing blanks in g of x.

00:42 So they told us g of x is an even function.

00:45 That means it's completely symmetrical on both sides.

00:47 So if f or g of negative 1 is 2, then g of 1 is also 2.

00:54 If g of negative 2 is 0, then g of positive 2 is also 0.

00:59 It's symmetrical on both sides.

01:00 And so now we can see if we can fill in f of g of x and we can come back to f of x later.

01:07 So let's see how much we can do of this bottom column.

01:10 So that's what this bottom row means.

01:12 When you see this zero, it looks like a zero.

01:15 It looks like it spells fog.

01:16 But this is really just a dot in between them.

01:19 That means so f of g of x means the same thing.

01:26 Another way to write this, which i think is a little easier to read, is f of g of x...

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