1. **Problem statement:** For each pair of functions, identify one characteristic they have in common and one characteristic that distinguishes them. 2. **Pair a) Functions:** $f(x) = \frac{1}{x}$ and $g(x) = x$ - **Common characteristic:** Both are defined for all real numbers except $x=0$ (domain excludes zero). - **Distinguishing characteristic:** $f(x)$ is a hyperbola with vertical and horizontal asymptotes, while $g(x)$ is a straight line with slope 1. 3. **Pair b) Functions:** $f(x) = \sin x$ and $g(x) = x$ - **Common characteristic:** Both are continuous functions for all real $x$. - **Distinguishing characteristic:** $\sin x$ is periodic with range $[-1,1]$, while $g(x)$ is non-periodic and unbounded. 4. **Pair 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, $g(x)$ is quadratic with a parabola shape and slope changes with $x$. 5. **Pair d) Functions:** $f(x) = 2^x$ and $g(x) = |x|$ - **Common characteristic:** Both are defined for all real numbers. - **Distinguishing characteristic:** $2^x$ is an exponential function increasing for all $x$, while $|x|$ is V-shaped and symmetric about the y-axis. This analysis helps understand similarities and differences in domain, range, shape, and behavior of these function pairs.