1. **Stating the problem:** We explore 5 functions focusing on aids in graph sketching: domain and range, intercepts, symmetries, asymptotes, and real-life applications. 2. **Function 1: $f(x) = x^2$** - Domain: All real numbers, $\mathbb{R}$ - Range: $[0, \infty)$ - Intercepts: $y$-intercept at $(0,0)$; $x$-intercept at $(0,0)$ - Symmetry: Even function, symmetric about y-axis. - Asymptotes: None. - Application: Projectile motion height vs time (ignoring air resistance). 3. **Function 2: $g(x) = \frac{1}{x}$** - Domain: $x \neq 0$ - Range: $y \neq 0$ - Intercepts: None. - Symmetry: Odd function, symmetric about origin. - Asymptotes: Vertical at $x=0$; horizontal at $y=0$. - Application: Electrical resistance vs conductance. 4. **Function 3: $h(x) = \sqrt{x}$** - Domain: $[0, \infty)$ - Range: $[0, \infty)$ - Intercepts: $(0,0)$ - Symmetry: None. - Asymptotes: None. - Application: Time for an object to fall a distance under gravity. 5. **Function 4: $p(x) = \ln(x)$** - Domain: $(0, \infty)$ - Range: $\mathbb{R}$ - Intercepts: $x$-intercept at $(1,0)$; no $y$-intercept. - Symmetry: None. - Asymptotes: Vertical at $x=0$. - Application: Population growth with continuous compounding. 6. **Function 5: $q(x)=e^x$** - Domain: $\mathbb{R}$ - Range: $(0, \infty)$ - Intercepts: $y$-intercept at $(0,1)$ - Symmetry: None. - Asymptotes: Horizontal at $y=0$. - Application: Radioactive decay and compound interest. Step-by-step solutions involve evaluating each property using algebraic or calculus methods. Graphical presentations show standard shapes: parabola, hyperbola, square root curve, logarithmic curve, exponential growth. Each concept is introduced simply: domain as permissible inputs, range as outputs, intercepts where graph meets axes, symmetry as geometric property, asymptotes as lines approached. These functions cover core graph-sketching techniques applicable widely.