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?

d. f² = ff

Accepted Answer

Welcome back, everyone. Let G be an odd function that is defined for all real numbers. Determine the nature, even odd, or neither of the function g cubed equals G multiplied by G multiplied by G. We given for answer choices A says even B odd, C neither even nor odd, and D cannot be determined. We know that she is an odd function, so we can basically say that if x is our independent variable, then she of of negative X. Must be equal to negative G of X. So this is the condition for an odd function, right? We are essentially evaluating G at negative X, and if the result is negative of G of X then The conclusion is that the function is odd. So in this case, we are looking at the function G cube. Of X, which is G of X, multiplied by G of X, multiplied by G of X. What if we evaluate G cubed of negative X to test for symmetry, right? We're essentially going to take G of negative X on the right hand side, multiplied by G of another negative X. Multiplied by g of negative X. So for every X we are now replacing it with negative X. And now what we want to do is simply understand that G of negative X is negative of G of X. So we can replace the right hand side with negative G of X multiplied by another negative G of X. Which is then multiplied by another negative G of X. And as we can see, we have -1 multiplied by -1, multiplied by -1. This gives us a negative sign overall, and then we are simply cubing G of X. So what can we see? Well, G cubed of negative X is equal to negative G cubed of X, which means that the function is odd. Because it means the criterion for an odd function and therefore our final answer to this problem corresponds to the answer choice B. 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?

d. f² = ff

Key Concepts

Even and Odd Functions

Function Composition and Products

Properties of Function Powers

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