1. **Problem statement:** Determine if the given functions are periodic, find the smallest period if it exists, and analyze properties such as evenness, oddness, and bijectivity. --- ### Part M1 230: Periodicity of functions from graphs 2. **Definition:** A function $f$ is periodic if there exists a smallest positive number $T$ such that $$f(x+T) = f(x)$$ for all $x$ in the domain. 3. **Analysis of graphs:** - a) The graph shows diagonal lines going downwards evenly spaced, which suggests a linear function, not periodic. - b) The jagged wave with peaks and troughs suggests a periodic function. The smallest period $T$ is the horizontal length after which the pattern repeats. - c) Steps going upwards diagonally indicate a non-periodic function. - d) Horizontal piecewise constant lines do not repeat periodically. 4. **Conclusion:** Only graph b) is periodic. The smallest period $T$ can be estimated by measuring the distance between repeating peaks or troughs. --- ### Part 231: Properties of functions 5. **Even function:** $f(-x) = f(x)$ 6. **Odd function:** $f(-x) = -f(x)$ 7. **Periodicity:** As above. 8. **From given graphs:** - a) Linear function, neither even nor odd, not periodic. - b) Function with values at discrete points, no periodicity. - c) Sinusoidal-like oscillations suggest periodicity, check symmetry for even/odd. - d) Discrete points, no periodicity. --- ### Part 237: Bijectivity and inverse functions 9. **Function a) $f(x) = 2x - 3$** - Linear, strictly increasing, hence bijective. - Inverse function: Solve $y = 2x - 3$ for $x$: $$y = 2x - 3 \Rightarrow 2x = y + 3 \Rightarrow x = \frac{y + 3}{2}$$ - So, $$f^{-1}(x) = \frac{x + 3}{2}$$ 10. **Function b) $f(x) = -2x + 2$** - Linear, strictly decreasing, hence bijective. - Inverse function: Solve $y = -2x + 2$ for $x$: $$y = -2x + 2 \Rightarrow -2x = y - 2 \Rightarrow x = \frac{2 - y}{2}$$ - So, $$f^{-1}(x) = \frac{2 - x}{2}$$ 11. **Graphical check:** The graphs of $f$ and $f^{-1}$ are symmetric about the line $y = x$ (the 1st median). This confirms the inverse relationship. --- **Summary:** - Only graph b) in M1 230 is periodic with smallest period $T$ estimated from the graph. - Functions in 231 mostly non-periodic except sinusoidal-like ones. - Functions in 237 are bijective with inverses as derived.