1. Problem 3: Determine if the graph defines $y$ as a function of $x$ for each graph. (a) The graph is an increasing curve passing through the origin resembling a cubic function. It passes the vertical line test (one $y$ for each $x$), so it defines $y$ as a function of $x$. (b) The graph is wave-like with a peak and trough crossing the $x$-axis twice. Since for some $x$ values, there are multiple $y$ values, it does not define $y$ as a function of $x$. (c) The graph is a vertically symmetric curve opening towards positive and negative $y$ resembling a sideways parabola. Vertical lines will intersect it in two points for some $x$, so it does not define $y$ as a function of $x$. (d) The graph is an ellipse centered at the origin. Vertical lines intersect the ellipse in two points, so it does not define $y$ as a function of $x$. 2. Problem 9: Find natural domain and range. (a) $f(x) = \frac{1}{x-3}$ - Domain: $x \neq 3$ - Range: $y \neq 0$ (b) $F(x) = \frac{x}{|x|}$ - Domain: $x \neq 0$ - Range: $\{-1, 1\}$ (c) $g(x) = \sqrt{x^2 - 3}$ - Domain: $x^2 - 3 \geq 0 \Rightarrow |x| \geq \sqrt{3}$ - Range: $y \geq 0$ (d) $G(x) = \sqrt{x^2 - 2x + 5}$ - Complete the square: $x^2 - 2x + 5 = (x-1)^2 + 4$ - Domain: all real numbers - Range: $y \geq 2$ (e) $h(x) = \frac{1}{1 - \sin x}$ - Domain: $1 - \sin x \neq 0 \Rightarrow \sin x \neq 1$ - Range: all real numbers except possibly $0$ (f) $H(x) = \sqrt{\frac{x^2 - 4}{x - 2}}$ - Simplify numerator: $x^2 - 4 = (x-2)(x+2)$ - Expression under root: $\frac{(x-2)(x+2)}{x-2} = x+2$ for $x \neq 2$ - Domain: $x \neq 2$, and $x+2 \geq 0 \Rightarrow x \geq -2$ - So domain is $[-2, 2) \cup (2, \infty)$ - Range: $y \geq 0$ 3. Problem 10: Find natural domain and range. (a) $f(x) = \sqrt{3 - x}$ - Domain: $3 - x \geq 0 \Rightarrow x \leq 3$ - Range: $y \geq 0$ (b) $F(x) = \sqrt{4 - x^2}$ - Domain: $4 - x^2 \geq 0 \Rightarrow |x| \leq 2$ - Range: $y \geq 0$ (c) $g(x) = 3 + \sqrt{x}$ - Domain: $x \geq 0$ - Range: $y \geq 3$ (d) $G(x) = x^3 + 2$ - Domain: all real numbers - Range: all real numbers (e) $h(x) = 3 \sin x$ - Domain: all real numbers - Range: $[-3, 3]$ (f) $H(x) = (\sin \sqrt{x})^{-2} = \frac{1}{\sin^2(\sqrt{x})}$ - Domain: $x \geq 0$, and $\sin(\sqrt{x}) \neq 0 \Rightarrow \sqrt{x} \neq k\pi$ for $k=0,1,2,\dots$ - Range: $(1, \infty)$ since $\sin^2$ is between 0 and 1, and reciprocal squared is at least 1 but undefined when $\sin \sqrt{x} =0$. Final answers compiled above.