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 entries in the table. $$\begin{array}{c|c|c|c|c|c}\hline x & -2 & -1 & 0 & 1 & 2 \\\hline f(x) & & & 0 & -2 & \\\hline g(x) & 0 & 2 & 1 & & \\ \hline(f \circ g)(x) & & 1 & -2 & & \\\hline\end{array}$$

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

Key Concepts

Odd Functions

An odd function is one that satisfies the property f(-x) = -f(x) for all x in its domain. This means that its graph is symmetric with respect to the origin, and knowing the value of f(x) for positive x allows you to determine the value for negative x by taking the negative of the corresponding positive value.

Even Functions

An even function satisfies the property g(-x) = g(x) for all x in its domain. This indicates that its graph is symmetric with respect to the y-axis, and knowing the value for a positive input immediately gives the same value for the corresponding negative input.

Function Composition

Function composition involves applying one function to the result of another, typically written as (f ? g)(x) = f(g(x)). In problems that require filling in table entries, understanding how the inner function g(x) is evaluated first and then used as the input for the outer function f(x) is crucial for determining the correct values.

Transcript

00:01 Okay, so we're looking to complete this table.

00:04 And so if f and g are functions, then the composition of the function of g and f is defined by, well, f composed g would be f of g of x, and g composed f would be g of x.

00:24 So we consider our table here.

00:28 And then from the table, we have that, well, f of zero, if f of x if x is zero then f of x is zero that means that f of zero is equal to zero so we have that f of zero is equal to zero we have that f of one is equal to negative two we have that g of negative two is equal to zero we have that g of negative one is equal to two we have that g of negative one is equal to two we have that g of zero is equal to one and in the composition, we have f -compose -g of negative 1.

01:09 So that's f of g of negative.

01:11 So you're taking g now and putting it into f.

01:13 And we have that f -of -g of negative 1 is equal to 1.

01:16 And that f -composed g of 0.

01:20 So f -composed g of 0 is equal to negative 2.

01:26 Okay.

01:29 So we suppose that f of x is an odd function and that g of x is an even function.

01:35 So in that case, we have that the fact that f is an odd function, that means that f of negative x is equal to negative f of x.

01:46 And if g is even, we have that g of negative x is equal to g of x.

01:54 That's a definition of an odd and even function.

01:56 Okay, so since f is odd, we know that f of negative 1 is equal to negative...