Final answer:
To determine symmetry of a function algebraically, we check for evenness, oddness, symmetry with respect to the x-axis, and symmetry with respect to y-axis.
Step-by-step explanation:
To determine the symmetry of a function algebraically, we need to check if it is an even function, an odd function, symmetric with respect to the x-axis, or symmetric with respect to the y-axis.
A) Even function:
A function is even if it satisfies the condition y(x) = y(-x). To determine if a function is even, we check if replacing x with -x in the function yields an equivalent expression.
B) Odd function:
A function is odd if it satisfies the condition y(x) = -y(-x). To determine if a function is odd, we check if replacing x with -x in the function yields the negative of the original expression.
C) Symmetric with respect to the x-axis:
A function is symmetric with respect to the x-axis if y(x) = -y(x). To determine this, we check if the function remains unchanged when we replace y with -y.
D) Symmetric with respect to the y-axis:
A function is symmetric with respect to the y-axis if y(x) = y(-x). To determine this, we check if the function remains unchanged when we replace x with -x.