1. For the pair $f(x) = \frac{1}{x}$ and $g(x) = x$: - Common characteristic: Both are functions that are defined for all $x \neq 0$ (domain excludes zero for $f(x)$), and both are real-valued functions. - Distinguishing characteristic: $f(x)$ is a hyperbola and is not a polynomial, it has a vertical asymptote at $x=0$, while $g(x)$ is a linear function with no asymptotes. 2. For the pair $f(x) = \sin x$ and $g(x) = x$: - Common characteristic: Both are continuous and differentiable functions for all real $x$. - Distinguishing characteristic: $f(x)$ is periodic with range $[-1,1]$, while $g(x)$ is not periodic and has an infinite range $(-\infty, \infty)$. 3. For the pair $f(x) = x$ and $g(x) = x^2$: - Common characteristic: Both are polynomial functions and both are continuous and differentiable everywhere. - Distinguishing characteristic: $f(x)$ is linear with constant slope 1, while $g(x)$ is quadratic and has a varying slope and a minimum point at $x=0$. 4. For the pair $f(x) = 2^x$ and $g(x) = |x|$: - Common characteristic: Both functions are continuous and non-negative for all real $x$. - Distinguishing characteristic: $2^x$ is an exponential function that grows without bound as $x \to \infty$, while $|x|$ is piecewise linear, symmetric about the $y$-axis, and grows linearly.