Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))

Verified step by step guidance

1. Identify the properties of even and odd functions: An even function satisfies f(x) = f(-x) for all x, and an odd function satisfies g(x) = -g(-x) for all x.
2. Since f is an even function and f(2) = 2, use the property of even functions to find f(-2).
3. Evaluate f(-2) using the property f(x) = f(-x).
4. Since g is an odd function and g(2) = -2, use the property of odd functions to find g(-2).
5. Evaluate f(g(2)) and g(f(-2)) using the values found for f(-2) and g(2).

Key Concepts

Even Functions

An even function is defined by the property that f(x) = f(-x) for all x in its domain. This means that the function's graph is symmetric with respect to the y-axis. For example, if f(2) = 2, then f(-2) must also equal 2, illustrating the even function's characteristic.

Odd Functions

An odd function satisfies the condition g(x) = -g(-x) for all x in its domain. This indicates that the function's graph is symmetric with respect to the origin. For instance, if g(2) = -2, then g(-2) must equal 2, demonstrating the odd function's defining property.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. For example, evaluating f(g(2)) means substituting g(2) into the function f. Understanding how to manipulate and evaluate compositions is crucial for solving problems involving multiple functions.