Assume f is an even function and g is an odd function. Use...

Assume $f$ is an even function and $g$ is an odd function. Use the (incomplete) table to evaluate the given compositions. $$\begin{array}{lrrrr}\hline x & 1 & 2 & 3 & 4 \\f(x) & 2 & -1 & 3 & -4 \\g(x) & -3 & -1 & -4 & -2 \\\hline\end{array}$$ a. $f(g(-1))$ b. $g(f(-4))$ c. $f(g(-3))$ d. $f(g(-2))$ e. $g(g(-1))$ f. $f(g(0)-1)$ g. $f(g(g(-2)))$ h. $g(f(f(-4)))$ i. $ g(g(g(-1)))$

Assume $f$ is an even function and $g$ is an odd function. Use the (incomplete) table to evaluate the given compositions.
$$\begin{array}{lrrrr}\hline x & 1 & 2 & 3 & 4 \\f(x) & 2 & -1 & 3 & -4 \\g(x) & -3 & -1 & -4 & -2 \\\hline\end{array}$$
a. $f(g(-1))$
b. $g(f(-4))$
c. $f(g(-3))$
d. $f(g(-2))$
e. $g(g(-1))$
f. $f(g(0)-1)$
g. $f(g(g(-2)))$
h. $g(f(f(-4)))$
i. $ g(g(g(-1)))$

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Key Concepts

Even Function

An even function is one that satisfies the property f(-x) = f(x) for all x in its domain. This symmetry about the y-axis is used to determine the value of the function for negative inputs using the values given for the corresponding positive inputs.

Odd Function

An odd function is characterized by the property g(-x) = -g(x) for all x in its domain. This means that the value at any negative argument is the negative of the value at the corresponding positive argument, a symmetry that is particularly useful for evaluating compositions that involve negative inputs.

Function Composition

Function composition involves applying one function to the result of another. For instance, in a composition like f(g(x)), you first evaluate g(x) and then substitute that result into f. Comprehending the order of operations is crucial for correctly solving compositions.

Table Lookup for Functions

Using a table to evaluate functions involves finding the function value corresponding to a specific input as listed in a provided table. When functions have defining properties (like even or odd), these properties help extend or infer values that may not be explicitly given in the table.

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