Even Function Analysis Of F(x), H(x), G(x), And S(x)

Hey guys! Let's dive into the world of functions and their symmetries. We've got a set of functions here: f(x) = sin(x^4 - x^2), h(x) = (|x| - 3)^3, g(x) = ln(|x|) + 3, and s(x) = sin^3(x). Our mission today is to figure out which of these functions are even. But before we jump into the specifics, let's quickly refresh our understanding of what even functions actually are. An even function is a function that satisfies the property f(x) = f(-x) for all x in its domain. Geometrically, this means that the graph of an even function is symmetric with respect to the y-axis. In simpler terms, if you fold the graph along the y-axis, the two halves will perfectly overlap. Now that we've got the basics covered, let's roll up our sleeves and get into the nitty-gritty of analyzing our functions. We'll take each function one by one and see if it fits the bill for being an even function. Ready to put on our mathematical hats and get started? Let's do this!

Analyzing h(x) = (|x| - 3)^3

Let's start by deep-diving into the function h(x) = (|x| - 3)^3. To determine if h(x) is even, we need to check if h(x) = h(-x) holds true for all x. So, the first step here is to calculate what h(-x) actually is. We do this by substituting -x into the function wherever we see x. This gives us h(-x) = (|-x| - 3)^3. Now, let's remember a key property of absolute values: the absolute value of a number is the same as the absolute value of its negative. In mathematical terms, |x| = |-x|. This is because the absolute value function gives us the distance of a number from zero, and distance is always a non-negative quantity. Applying this property to our expression for h(-x), we get h(-x) = (|x| - 3)^3. Now, compare this with our original function, h(x) = (|x| - 3)^3. Lo and behold, they are exactly the same! This means that we have successfully shown that h(x) = h(-x) for all x. Therefore, we can confidently conclude that the function h(x) = (|x| - 3)^3 is indeed an even function. This is a significant finding, and it tells us a lot about the symmetry of this function. The graph of h(x) will be symmetric about the y-axis, which can be a useful piece of information when we're trying to visualize or sketch the function. But we're not done yet! We still have other functions to investigate, so let's keep the momentum going and move on to our next candidate.

Diving into g(x) = ln(|x|) + 3

Now, let's turn our attention to the function g(x) = ln(|x|) + 3. Just like with h(x), our goal here is to determine if g(x) is an even function. Remember, for a function to be even, it must satisfy the condition g(x) = g(-x) for all x in its domain. So, our first step is to find g(-x). We do this by replacing every instance of x in the function with -x. This gives us g(-x) = ln(|-x|) + 3. Now, let's bring back that handy property of absolute values we talked about earlier: |x| = |-x|. Applying this to our expression for g(-x), we get g(-x) = ln(|x|) + 3. Take a good look at this. Does it look familiar? It should! It's exactly the same as our original function, g(x) = ln(|x|) + 3. This means we've successfully shown that g(x) = g(-x) for all x. And what does that tell us? It tells us that the function g(x) = ln(|x|) + 3 is, without a doubt, an even function. This is another important result! It means that the graph of g(x) is also symmetric with respect to the y-axis. When you're working with functions, recognizing these symmetries can be a real game-changer. It can simplify graphing, help you solve equations, and give you a deeper understanding of the function's behavior. But we're not stopping here. We've still got two more functions to analyze, so let's keep going and see what we discover.

Examining f(x) = sin(x^4 - x^2)

Alright, let's shift our focus to the function f(x) = sin(x^4 - x^2). As we've done with the previous functions, we need to determine if f(x) is even by checking if f(x) = f(-x) holds true. To do this, we'll start by finding f(-x). We substitute -x into the function wherever we see x, which gives us f(-x) = sin((-x)^4 - (-x)^2). Now, let's simplify this expression. Remember that raising a negative number to an even power results in a positive number. So, (-x)^4 = x^4 and (-x)^2 = x^2. Substituting these back into our expression, we get f(-x) = sin(x^4 - x^2). Take a close look at this. It's the same as our original function, f(x) = sin(x^4 - x^2). This means we've shown that f(x) = f(-x), which confirms that f(x) is indeed an even function! So, we've added another function to our list of even functions. This is pretty cool because it shows how different types of functions (like trigonometric functions in this case) can exhibit symmetry. The fact that f(x) is even tells us that its graph will be symmetric about the y-axis, just like the previous even functions we've analyzed. This kind of symmetry can be super helpful when we're trying to visualize the function or understand its behavior. But we've still got one more function to go, so let's keep our analytical hats on and move on to the final function in our set.

Investigating s(x) = sin^3(x)

Last but not least, let's investigate the function s(x) = sin^3(x). As with the other functions, we want to determine if s(x) is even by checking if s(x) = s(-x). To do this, we'll first find s(-x). Substituting -x for x in the function gives us s(-x) = sin^3(-x). Now, let's recall a key property of the sine function: sin(-x) = -sin(x). This property tells us that the sine function is an odd function. Applying this to our expression, we get s(-x) = (-sin(x))^3. When we cube a negative quantity, the result is also negative. Therefore, (-sin(x))^3 = -sin^3(x). So, we have s(-x) = -sin^3(x). Now, let's compare this to our original function, s(x) = sin^3(x). Notice that s(-x) = -s(x), which means that s(x) is an odd function, not an even function. This is a crucial distinction! Odd functions have a different type of symmetry compared to even functions. Odd functions are symmetric about the origin, meaning that if you rotate the graph 180 degrees about the origin, it will look the same. So, the function s(x) = sin^3(x) doesn't fit the bill for being an even function. This highlights the importance of carefully checking the condition f(x) = f(-x) (for even functions) or f(-x) = -f(x) (for odd functions) to correctly classify a function's symmetry. We've now analyzed all four functions, so let's take a step back and summarize our findings.

Conclusion

Okay, guys, we've done some serious function analysis today! We started with four functions: f(x) = sin(x^4 - x^2), h(x) = (|x| - 3)^3, g(x) = ln(|x|) + 3, and s(x) = sin^3(x). Our mission was to determine which of these functions are even. We carefully examined each function, applying the definition of even functions: a function f(x) is even if f(x) = f(-x) for all x in its domain. Through our analysis, we discovered that h(x) = (|x| - 3)^3 and g(x) = ln(|x|) + 3 are indeed even functions. We also found that f(x) = sin(x^4 - x^2) is even. However, we found that s(x) = sin^3(x) is an odd function, not an even function, because it satisfies the condition s(-x) = -s(x). This exercise has not only helped us identify even functions but has also reinforced our understanding of function symmetry in general. Recognizing whether a function is even or odd can provide valuable insights into its graphical behavior and properties, making it a crucial tool in mathematical analysis. So, the final answer is that h(x) and g(x) are even functions. We've successfully navigated the world of function symmetry, and hopefully, you've gained a clearer understanding of even and odd functions along the way. Keep exploring the fascinating world of mathematics, and remember to always question, analyze, and have fun with it! Cheers!