Use the table below to determine if
is an even function, an odd function, or neither. Justify your reasoning. x f(x) -3 -7 -2 3 -1 2 0 0 1 -2 2 -3 3 7
The Correct Answer and Explanation is:
The function represented by the given table is an odd function. This classification is based on the symmetry of the function’s values about the origin.
To determine whether a function is even, odd, or neither, we analyze the relationship between f(x)f(x) and f(−x)f(-x). An even function satisfies the property f(x)=f(−x)f(x) = f(-x) for all x in the domain. This means the graph is symmetric with respect to the y-axis. An odd function satisfies the condition f(−x)=−f(x)f(-x) = -f(x), meaning it is symmetric about the origin.
Let us evaluate the values from the table:
- For x=−3x = -3, f(−3)=−7f(-3) = -7 and f(3)=7f(3) = 7. Here, f(3)=−f(−3)f(3) = -f(-3)
- For x=−2x = -2, f(−2)=3f(-2) = 3 and f(2)=−3f(2) = -3. Hence, f(2)=−f(−2)f(2) = -f(-2)
- For x=−1x = -1, f(−1)=2f(-1) = 2 and f(1)=−2f(1) = -2. Again, f(1)=−f(−1)f(1) = -f(-1)
- For x=0x = 0, f(0)=0f(0) = 0, which satisfies both even and odd conditions since f(0)=−f(0)=0f(0) = -f(0) = 0
Every x-value has a corresponding opposite, and each pair follows the rule f(−x)=−f(x)f(-x) = -f(x). This means the graph, if plotted, would exhibit rotational symmetry about the origin. That symmetry is a hallmark of an odd function.
In conclusion, the function is odd because each pair of symmetric inputs yields outputs that are negatives of each other. This pattern remains consistent across all values provided, so the function meets the strict definition of oddness through origin symmetry.
