Discuss any symmetry present in the function.
Check for even or odd symmetry and explain how to verify it.
To determine if a function has even or odd symmetry, we need to analyze its symmetry properties in relation to its axis of symmetry.
1. Even Symmetry:
A function is said to exhibit even symmetry if it is symmetric about the y-axis, meaning that for any x-value, f(x) = f(-x). This implies that the graph of the function is identical on both sides of the y-axis.
To verify even symmetry, we can substitute -x into the function and check if it yields the same function. If f(x) = f(-x), then the function is even symmetric.
For example, let's consider the function f(x) = x^2. Substituting -x, we have f(-x) = (-x)^2 = x^2, which is the same as f(x). Hence, the function f(x) = x^2 is even symmetric.
2. Odd Symmetry:
A function is said to exhibit odd symmetry if it is symmetric about the origin (0,0), meaning that for any x-value, f(x) = -f(-x). This implies that the graph of the function is rotated 180 degrees around the origin, producing the same shape but reversed on the opposite side of the origin.
To verify odd symmetry, we can substitute -x and verify if it produces the negative of the function. If f(x) = -f(-x), then the function is odd symmetric.
For example, let's consider the function f(x) = x^3. Substituting -x, we have f(-x) = (-x)^3 = -x^3, which is the negative of f(x). Hence, the function f(x) = x^3 is odd symmetric.
In summary, to check for even symmetry, substitute -x into the function and verify if it remains unchanged (f(x) = f(-x)). For odd symmetry, substitute -x and verify if it produces the negative of the function (f(x) = -f(-x)).