1. **Problem Statement:** Determine periodicity, smallest period if exists, evenness, oddness, and bijectivity of given functions from graphs and formulas. 2. **Periodicity Definition:** A function $f$ is periodic if there exists smallest positive $T$ such that $$f(x+T) = f(x) \text{ for all } x.$$ The smallest such $T$ is called the fundamental period. 3. **Analysis of Graphs (M1 230):** - a) Diagonal lines downward evenly spaced indicate a linear function, which is not periodic. - b) Jagged wave with repeating peaks/troughs is periodic. The smallest period $T$ equals the horizontal distance between repeating features. - c) Steps going diagonally upward are non-periodic. - d) Horizontal piecewise constant lines do not repeat periodically. 4. **Conclusion on Periodicity:** Only graph b) is periodic with smallest period $T$ estimated from the graph. 5. **Even and Odd Functions (M1 231):** - Even: $f(-x) = f(x)$ - Odd: $f(-x) = -f(x)$ 6. **From given graphs:** - a) Linear function, neither even nor odd, not periodic. - b) Discrete points, no periodicity. - c) Sinusoidal-like oscillations suggest periodicity; check symmetry for even/odd. - d) Discrete points, no periodicity. 7. **Bijectivity and Inverse Functions (M1 237):** - a) $f(x) = 2x - 3$ is linear, strictly increasing, hence bijective. Solve for inverse: $$y = 2x - 3 \Rightarrow 2x = y + 3 \Rightarrow x = \frac{y + 3}{2}$$ So, $$f^{-1}(x) = \frac{x + 3}{2}$$ - b) $f(x) = -2x + 2$ is linear, strictly decreasing, hence bijective. Solve for inverse: $$y = -2x + 2 \Rightarrow -2x = y - 2 \Rightarrow x = \frac{2 - y}{2}$$ So, $$f^{-1}(x) = \frac{2 - x}{2}$$ 8. **Graphical Check:** Graphs of $f$ and $f^{-1}$ are symmetric about the line $y = x$, confirming inverse relationship. **Summary:** - Only graph b) in M1 230 is periodic with smallest period $T$. - Functions in M1 231 mostly non-periodic except sinusoidal-like ones. - Functions in M1 237 are bijective with inverses as derived.