Question

select the correct answer from each drop-down menu. complete the statements about the graph of a function. when a graph is symmetric about the origin, it is an function. when a graph is symmetric about the y-axis, it is an function. a function is considered if $f(-x) = f(x)$, and it is considered if $f(-x) = -f(x)$. reset next

Explanation:

Step1: Recall Odd Function Definition

An odd function satisfies \( f(-x) = -f(x) \) and its graph is symmetric about the origin. So the first blank (symmetric about origin) is "odd".

Step2: Recall Even Function Definition

An even function satisfies \( f(-x) = f(x) \) and its graph is symmetric about the \( y \)-axis. So the second blank (symmetric about \( y \)-axis) is "even".

Step3: Match \( f(-x) = f(x) \)

The equation \( f(-x) = f(x) \) is the definition of an even function, so the third blank is "even".

Step4: Match \( f(-x) = -f(x) \)

The equation \( f(-x) = -f(x) \) is the definition of an odd function, so the fourth blank is "odd".

Answer:

When a graph is symmetric about the origin, it is an \(\boldsymbol{\text{odd}}\) function.
When a graph is symmetric about the \( y \)-axis, it is an \(\boldsymbol{\text{even}}\) function.
A function is considered \(\boldsymbol{\text{even}}\) if \( f(-x) = f(x) \), and it is considered \(\boldsymbol{\text{odd}}\) if \( f(-x) = -f(x) \).