1. **Problem:** For each pair of functions, identify one characteristic they share and one characteristic that distinguishes them. 2. **Part a) Functions:** $f(x)=\frac{1}{x}$ and $g(x)=x$ - Common characteristic: Both functions are defined for all $x \neq 0$ (excluding zero for $f(x)$). - Distinguishing characteristic: $f(x)$ is a rational function with a vertical asymptote at $x=0$, while $g(x)$ is a linear function with no asymptotes. 3. **Part b) Functions:** $f(x)=\sin x$ and $g(x)=x$ - Common characteristic: Both functions are continuous and differentiable for all real $x$. - Distinguishing characteristic: $f(x)$ is periodic with range $[-1,1]$, while $g(x)$ is unbounded and strictly increasing. 4. **Part 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 with constant slope 1, while $g(x)$ is quadratic with slope changing linearly and is always nonnegative. 5. **Part d) Functions:** $f(x)=2^x$ and $g(x)=|x|$ - Common characteristic: Both are nonnegative for all real $x$. - Distinguishing characteristic: $f(x)$ is an exponential function increasing for all $x$, while $g(x)$ is symmetric about the y-axis and grows linearly for $x>0$ and $x<0$. **Final answers:** - a) Common: domain excludes 0; Distinct: $1/x$ has asymptote, line does not. - b) Common: continuous & differentiable everywhere; Distinct: sine is periodic and bounded, $x$ is unbounded and linear. - c) Common: pass through origin; Distinct: linear vs quadratic with different curvature. - d) Common: nonnegative outputs; Distinct: exponential growth vs absolute value (piecewise linear and symmetric).