Combining Functions

Assume t...

Question

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

g. g ∘ f

Accepted Answer

Welcome back, everyone. Let A be an even function and B an odd function, which are both defined for all real numbers. What is the nature of the function A of B? A even B odd, C neither even nor odd, and D cannot be determined. So essentially let's analyze each function. We know that a of X, let's suppose that our independent variable is X. It is an even function because it is an even function, we can say that if we calculate a of negative X, then the result will be the original function A of X. So this is the condition for an even function. Now, on the other hand, B. Of X is an odd function, and this means that if we evaluate b of negative X, this would be identical to negative b of X, the condition for an odd function, and now we want to evaluate the composition a composition b. Let's suppose that our independent variable is X. So it basically means that we're evaluating A of B of X, right? Now what is the nature of this function? Well, we basically need to substitute negative X. And evaluate the results. We're going to get A B of negative X. We know that B of negative X corresponds to negative B of X, so we're going to get E of negative B of X. And now, as we know, A of negative axis, A of X, the main difference here is that our X is now replaced with B of X, but it makes no difference because we're applying the same property of the. A of X, which is an even function, right? So it has an internal plan of symmetry. So this means that the final answer is A of B X. Because A is an even function, so we can eliminate that negative sign A of negative axis, A X. So A of negative B of X is equal to A of positiveb of x. What can we see from here? Well, we started with a of B X. Right, which was. The original function and then separately we evaluated a of B of negative X. So we can say that a of B of negative X is equal to A B X, which essentially implies that a composition B is an even function, right, because it meets the criterion of. A negative X being equal to A of X, the main difference is our independent variable which simply becomes a function B of X. So our final answer to this problem is option A even. Thank you for watching.

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

g. g ∘ f

Key Concepts

Even and Odd Functions

Function Composition

Properties of Composed Functions

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