Composition ​of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>


i. g(g(g(-1)))

Verified step by step guidance

1. Identify the property of odd functions: For an odd function $g(x)$, $g(-x)=-g(x)$.
2. Apply the odd function property to simplify $g(-1)$ to $-g(1)$.
3. Evaluate $g(g(-1))$ by substituting $g(-1)=-g(1)$ into the function, resulting in $g(-g(1))$.
4. Use the odd function property again to simplify $g(-g(1))$ to $-g(g(1))$.
5. Finally, evaluate $g(g(g(-1)))$ by substituting $g(g(-1))=-g(g(1))$ into the function, resulting in $g(-g(g(1)))$, and simplify using the odd function property to $-g(g(g(1)))$.

Key Concepts

Even and Odd Functions

Even functions are symmetric about the y-axis, meaning that f(x) = f(-x) for all x in their domain. Odd functions, on the other hand, are symmetric about the origin, satisfying the condition g(x) = -g(-x). Understanding these properties is crucial for evaluating compositions of such functions, as they dictate how the function values behave under negation.

Function Composition

Function composition involves applying one function to the result of another. If you have two functions f and g, the composition g(f(x)) means you first apply f to x, then apply g to the result. This concept is essential for evaluating expressions like g(g(g(-1))) as it requires sequentially substituting the output of one function into the next.

Evaluating Functions at Specific Points

To evaluate a function at a specific point, you substitute that point into the function's expression. For example, to find g(-1), you would look up the value of g at -1 in the provided table. This step is necessary for calculating compositions, as each function's output becomes the input for the next function in the sequence.