Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check

step 1 Hi guys. Today we're gonna be doing problem number 59. We were given the function Ethel X, and y

Determine whether f is even, odd, or neither. If you have a...

Key Concepts

Even Function

An even function is one that satisfies the condition f(x) = f(-x) for all x in its domain, meaning its graph is symmetric with respect to the y-axis. This property is useful when analyzing the symmetry and behavior of functions.

Odd Function

An odd function satisfies the condition f(-x) = -f(x) for all x in its domain, which indicates that its graph is symmetric about the origin. Recognizing odd functions can help in simplifying computations and understanding integral properties over symmetric intervals.

Determining Function Parity

To determine whether a function is even, odd, or neither, one typically substitutes -x into the function and compares the result with f(x) and -f(x). This algebraic method is complemented by graphical analysis, where the symmetry of the function's graph is examined to confirm its parity.

Transcript

00:02 Hi guys.

00:03 Today we're gonna be doing problem number 59.

00:05 We were given the function ethel x, and you have to determine whether it is odd, even or neither.

00:11 So the test you're gonna have to do is you're going to need to plug in negative x into the function f of x.

00:18 And so if you obtain the result, um, negative f of x, that means that your function is odd and you can see through.

00:25 During this example, we find that x of the third is action, not function.

00:30 And so if we were to graft dysfunction, we would see that their symmetry about the origin.

00:35 However, if we plug in negative accent to f of x and we obtain ffx again, which we can see with this cortical function x to the fourth, then we know that the function is actually even and their symmetry about the y axis.

00:50 Okay, so now we're going to be actually solving this problem.

00:54 So i'm going to be typing off screen so that okay, so the first step of this problems, we're going to need to be plugging in negative x into the equation, so that would look like this right here.

01:29 Okay, so all i did was plug in negative x where x should be here.

01:34 So now the next step here is we're going to need to do a little simplification.

01:42 So what that means is negative x is going to stay the same.

01:46 However, whenever you square negative x, this is gonna be equal to x squared.

01:55 So that would give you this expression right here.

02:03 So just a little simplification is all...

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