1. Problem: For each pair of functions, identify one characteristic they have in common and one that distinguishes them. 2. a) Functions: $f(x) = \frac{1}{x}$ and $g(x) = x$ - Common characteristic: Both are functions defined for all $x \neq 0$ (excluding zero where $f$ is undefined). - Distinguishing characteristic: $f(x)$ is a reciprocal function (hyperbola shape, decreasing on positive $x$), and $g(x)$ is a linear function increasing steadily. 3. b) Functions: $f(x) = \sin x$ and $g(x) = x$ - Common characteristic: Both are continuous everywhere. - Distinguishing characteristic: $f(x)$ is periodic with period $2\pi$, oscillating between $-1$ and $1$, while $g(x)$ is not periodic and unbounded. 4. c) Functions: $f(x) = x$ and $g(x) = x^2$ - Common characteristic: Both pass through the origin $(0,0)$. - Distinguishing characteristic: $f(x)$ is linear, increasing everywhere; $g(x)$ is a parabola, always non-negative and symmetric about the y-axis. 5. d) Functions: $f(x) = 2^x$ and $g(x) = |x|$ - Common characteristic: Both are defined for all real $x$ and both produce non-negative outputs. - Distinguishing characteristic: $f(x)$ is exponential and continuous, increasing for all $x$, while $g(x)$ is piecewise linear and symmetric about the y-axis. Final answer: See the detailed characteristics above for each pair.