Properties of Functions: Even, Odd, or Neither

Q1. Determine whether the functions P, Q, and R (shown in the graphs below) are even, odd, or neither.

Background

Topic: Symmetry of Functions (Even and Odd Functions)

This question tests your understanding of how to determine if a function is even, odd, or neither by analyzing its graph. Recognizing these properties is important in calculus for simplifying integrals and understanding function behavior.

Key Terms and Formulas

• Even Function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain. Its graph is symmetric about the y-axis.

• Odd Function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain. Its graph is symmetric about the origin.

• Neither: If a function does not satisfy either condition, it is neither even nor odd.

Step-by-Step Guidance

1. Examine the graph of function P. Check if the left and right sides of the y-axis are mirror images (y-axis symmetry) or if rotating the graph 180° about the origin gives the same graph (origin symmetry).

2. Repeat the symmetry check for function Q. Look for y-axis symmetry (even) or origin symmetry (odd).

3. Analyze the graph of function R in the same way, checking for both types of symmetry.

4. For each function, compare $f(x)$ and $f(-x)$ visually using the graph. If the graph is unchanged when reflected over the y-axis, it is even. If it is unchanged when rotated 180° about the origin, it is odd.

5. Decide for each function whether it is even, odd, or neither based on your symmetry observations. Write down your reasoning for each.

Try solving on your own before revealing the answer!

Final Answer:

• P: Even function. The graph of P is symmetric about the y-axis, so $P(-x) = P(x)$.

• Q: Even function. The graph of Q is also symmetric about the y-axis, so $Q(-x) = Q(x)$.

• R: Odd function. The graph of R is symmetric about the origin, so $R(-x) = -R(x)$.

Even functions have y-axis symmetry, and odd functions have origin symmetry. By visually inspecting the graphs, you can determine the symmetry type for each function.