Q. 3 - Calculus

Question

Fill in the blanks: The graph of every odd function is symmetric about ______. The graph of every even function is symmetric about ______.

Step-by-Step Solution

Answer

On filling up the blanks, we get, "The graph of every odd function is symmetric about the origin. The graph of every even function is symmetric about the y-axis."

Step 1. Explanation of the functions.

A function f  is an odd function if $f(-x) = -f\left(x\right)$ for all x in the domain of f.

A function f  is an even function if $f(-x) = f\left(x\right)$ for all x in the domain of f.

Step 2. Explain the graph of the functions.

The graph of an odd function has symmetry with respect to the origin, which means its graph remains unchanged after rotation of $180$ degrees about the origin.

The graph of an even function is symmetric with respect to the y-axis, which means its graph remains unchanged after reflection about the y-axis.

On completing the sentence, we get, "The graph of every odd function is symmetric about the origin. The graph of every even function is symmetric about the y-axis."