Determine whether the function f(x) = x³ - 3x is even, odd, or neither

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To determine whether the function f of x equals x cubed minus 3x is even, odd, or neither, we need to check how the function behaves when we replace x with negative x. Let's start by looking at the graph of the function. Notice that the function passes through the origin, and the graph appears to have rotational symmetry about the origin. This is a visual clue that the function might be odd. To determine if the function is odd, we need to check if f of negative x equals negative f of x. Let's substitute negative x into our function. f of negative x equals negative x cubed minus 3 times negative x. Simplifying, we get negative x cubed plus 3x. Now, let's find negative f of x. Negative f of x equals negative times x cubed minus 3x. Distributing the negative sign, we get negative x cubed plus 3x. Since f of negative x equals negative f of x, we can conclude that the function is odd. Let's visualize the symmetry of our odd function. An odd function has rotational symmetry about the origin. This means if we take any point on the graph and rotate it 180 degrees around the origin, we get another point on the graph. For example, if the point (2, 2) is on the graph, then the point (-2, -2) must also be on the graph. This is a key property of odd functions. Other properties include that f of negative x equals negative f of x for all x in the domain, and that odd functions contain only odd powers of x or terms with odd symmetry. Our function f(x) equals x cubed minus 3x consists of x cubed, which is an odd power, and negative 3x, which is also odd. Therefore, f(x) equals x cubed minus 3x is indeed an odd function. Let's compare even and odd functions to better understand the difference. An even function satisfies the condition f of negative x equals f of x. A classic example is g of x equals x squared. Even functions are symmetric about the y-axis, meaning if you fold the graph along the y-axis, the two halves match perfectly. On the other hand, an odd function satisfies the condition f of negative x equals negative f of x, which is what we proved for our function f of x equals x cubed minus 3x. Odd functions have rotational symmetry about the origin, meaning if you rotate the graph 180 degrees around the origin, it maps onto itself. Looking at both graphs, we can clearly see these different symmetry properties. The blue parabola g of x equals x squared is symmetric about the y-axis, while our red cubic function f of x equals x cubed minus 3x is symmetric about the origin. Therefore, we confirm that our function is odd. Let's summarize what we've learned. To determine whether a function is even, odd, or neither, we need to check specific conditions. For a function to be odd, we must verify that f of negative x equals negative f of x for all x in the domain. For our function f of x equals x cubed minus 3x, we calculated f of negative x, which equals negative x cubed plus 3x. We also calculated negative f of x, which equals negative x cubed plus 3x. Since these expressions are identical, we confirmed that f of negative x equals negative f of x, proving that our function is odd. Odd functions have rotational symmetry about the origin and typically contain only odd powers of x or terms with odd symmetry. Our function consists of x cubed, which has an odd power, and negative 3x, which also has odd symmetry. Therefore, we can confidently conclude that the function f of x equals x cubed minus 3x is an odd function.