Exercises 107–109 will help you prepare for the material covered ...

Question

Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x^4−2x^2+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

Accepted Answer

Welcome back. I'm so glad you're here. We have a function F of X equals four X to the fourth minus seven X squared plus one. And we are asked to determine if this function is even odd or neither. Will we recall from previous lessons that to determine if a function is even odd or neither. We are going to substitute a negative X into our half of x function. And there are three possible outcomes. If we substitute a negative X in and we get back our original equation or in other words, none of the signs change on any of our terms. Then this is by definition and even function if we substitute a negative acts in for all of our X terms in our original F of X function and we get back a negative F of X, meaning that every single terms sign has changed. Then that by definition is an odd function. And if we substitute a negative X in for every X in our original f of X function and it doesn't come back with our original function and it doesn't come back with a negative affects meaning not every sign has changed but some of them have then this is neither even nor odd. Those are three possible outcomes. So let's go ahead and substitute a negative X. Anywhere where we had an X in our original half of X function. So for F of negative X. This will be four times negative X to the fourth power minus seven times negative X squared plus one. Now to do the simplification, we look first at this negative X to the fourth power negative X to the fourth is going to be X to the fourth. Bring down that four knit and then minus seven times negative X squared. Well negative X squared is positive X squared. Then bring down plus one. Well, we've simplified it and this is the same as our original affects equation, meaning that this is an even function. So now we just have to describe the symmetry. Will we recall from previous lessons that by definition and even function is symmetrical about the Y axis. We look at our answer choices and this matches with answer choice B. Well done. We'll catch on the next one.

Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x^4−2x^2+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

Key Concepts

Even and Odd Functions

Graphical Symmetry

Polynomial Functions

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