Question

question 13 (1 point)

the point \\((-5, -5)\\) is on a graph of an odd function. what other point(s) must be on the graph of the function?

\\((5, 5)\\)

\\((5, -5)\\)

\\((0, 0)\\) & \\((-5, 5)\\)

\\((0, 0)\\) & \\((5, 5)\\)

\\((0, 0)\\) & \\((5, -5)\\)

\\((0, 0)\\)

Explanation:

🆕 New Concept Discovered: Even and Odd Functions
Symmetry about the origin vs. the y-axis

Step 1: Definition of an Odd Function

An odd function \( f(x) \) satisfies the algebraic condition:

$$ f(-x) = -f(x) $$

In terms of coordinate points, if a point \( (x, y) \) lies on the graph of an odd function, then the point \( (-x, -y) \) must also lie on the graph. This represents a reflection through the origin.

Step 2: Apply the Rule to the Given Point

We are given that the point \( (-5, -5) \) is on the graph. Here, \( x = -5 \) and \( y = -5 \).

Applying the odd function rule:

$$ (-x, -y) = (-(-5), -(-5)) = (5, 5) $$

Therefore, the point \( (5, 5) \) must also be on the graph.

Step 3: Evaluate the Origin (0,0)

An odd function is only guaranteed to pass through the origin \( (0, 0) \) if \( x = 0 \) is in the domain of the function. Since we are not given any information about the domain of this function, we cannot assume that \( (0, 0) \) must be on the graph.

Thus, the only point we can be absolutely certain is on the graph is \( (5, 5) \).

Answer:

\( (5, 5) \)