In Exercises 5–8, determine whether the graph of the function ...

Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x²/⁵

Accepted Answer

Welcome back everyone to another video. For the given function below is a symmetric about the y axis, the origin, both or neither. We're given Y equals X to the power of 4 divided by 7. For this problem, we want to follow simple steps whenever we want to decide if a function has some sort of symmetry. So let's test for symmetry. The first step is to define F of X, the original function. The second step is to define F of negative X and then see if F of negative X is equal to the original function F of X. If that's the case, we have an even function, and by definition, an even function would be symmetric relative to the y axis. Alternatively, if f of negative X is equal to negative F of X. Then we have an odd function. Which is symmetric relative to the origin. If none of these two conditions are met, we can say that it's neither, right? So now we're going to look at the function, which is F of X equals x the power of 4 divided by 7. And as we can see, our next step is to validate F of negative X. So X to the power of 4 becomes negative X or raise to the power of 4. And now we're dividing by 7. Because we are raising. Negative X to an even power we can essentially say that the result will be a positive number so we get we can get rid of the negative sign, and we can write our numerator as positive X to the power of 4 divided by 7. By a simple observation we can see that this expression is the same as the original function. So now we can see that F of negative X. is equal to F of X and this corresponds to the first condition, right? So if F of negative X in this case is equal to F of X. Then we can conclude that our function is even. And if it is even, it is symmetric about the y axis which is our final answer to this problem. Thank you for watching.

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x²/⁵

Key Concepts

Symmetry about the y-axis

Symmetry about the origin

Analyzing function symmetry

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