1. **Classify each function as odd, even, or neither.** We use the definitions: - A function $f$ is **even** if $f(-x) = f(x)$ for all $x$. - A function $f$ is **odd** if $f(-x) = -f(x)$ for all $x$. - Otherwise, it is **neither**. --- **a.** $f(x) = 3x^4 + 3$ 1. Compute $f(-x)$: $$f(-x) = 3(-x)^4 + 3 = 3x^4 + 3 = f(x)$$ 2. Since $f(-x) = f(x)$, $f$ is **even**. --- **b.** $f(x) = x^5 - 4x$ 1. Compute $f(-x)$: $$f(-x) = (-x)^5 - 4(-x) = -x^5 + 4x = -(x^5 - 4x) = -f(x)$$ 2. Since $f(-x) = -f(x)$, $f$ is **odd**. --- **c.** $f(x) = \frac{1}{x^2 + 1}$ 1. Compute $f(-x)$: $$f(-x) = \frac{1}{(-x)^2 + 1} = \frac{1}{x^2 + 1} = f(x)$$ 2. Since $f(-x) = f(x)$, $f$ is **even**. --- **d.** $f(x) = x^2 + x - 3$ 1. Compute $f(-x)$: $$f(-x) = (-x)^2 + (-x) - 3 = x^2 - x - 3$$ 2. Compare $f(-x)$ and $f(x)$: - $f(-x) \neq f(x)$ - $f(-x) \neq -f(x)$ 3. So, $f$ is **neither** even nor odd. --- **Final classifications:** - a: even - b: odd - c: even - d: neither 2. **Graph transformations for even and odd extensions of $f$ defined on $0 \leq x \leq 5$.** - For an **even** function $g$, define $g(x) = f(|x|)$, so the graph is symmetric about the $y$-axis. - For an **odd** function $h$, define $h(x) = \begin{cases} f(x), & x \geq 0 \\ -f(-x), & x < 0 \end{cases}$, so the graph is symmetric about the origin. (Sketches would show these symmetries.) 3. **If $f(x) = mx + b$ is even, find $m$ and $b$.** 1. Since $f$ is even, $f(-x) = f(x)$: $$m(-x) + b = mx + b$$ 2. Simplify: $$-mx + b = mx + b$$ 3. Subtract $b$ from both sides: $$-mx = mx$$ 4. Add $mx$ to both sides: $$0 = 2mx$$ 5. For all $x$, this implies: $$m = 0$$ 6. So $f(x) = b$, a constant function. 4. **Identify graphs of $y = f(x)$ and $y = f(x - 1)$.** - $y = f(x - 1)$ is the graph of $f$ shifted **right** by 1 unit. - The graph shifted right by 1 corresponds to $y = f(x - 1)$. 5. **Write expressions for two graphs in terms of $f(x)$.** - The graph shifted left by 1 is $y = f(x + 1)$. - The graph shifted right by 1 is $y = f(x - 1)$. **Summary:** - Problem 1: classification of functions. - Problem 2: graph symmetry for even and odd extensions. - Problem 3: linear function evenness implies $m=0$. - Problem 4: horizontal shift identification. - Problem 5: expressions for shifted graphs.