Analyze the given functions: f1(x) = x, f2(x) = x^2, f3(x) = √x, f4(x) = x^3, f5(x) = -x - 2, f6(x) = x + 2.

Steps to Solve

1. Identify the Functions and Their Forms

   The functions provided are:

   • $f_1(x) = x$ (linear function)
   • $f_2(x) = x^2$ (quadratic function)
   • $f_3(x) = \sqrt{x}$ (square root function)
   • $f_4(x) = x^3$ (cubic function)
   • $f_5(x) = -x - 2$ (linear function)
   • $f_6(x) = x + 2$ (linear function)

2. Analyze Each Function's Characteristics

   Evaluate the properties including:

   • Domain and Range:
     • $f_1(x)$: Domain: all real numbers; Range: all real numbers.
     • $f_2(x)$: Domain: all real numbers; Range: $[0, \infty)$.
     • $f_3(x)$: Domain: $[0, \infty)$; Range: $[0, \infty)$.
     • $f_4(x)$: Domain: all real numbers; Range: all real numbers.
     • $f_5(x)$: Domain: all real numbers; Range: all real numbers.
     • $f_6(x)$: Domain: all real numbers; Range: all real numbers.

3. Determine Continuity and Behavior

   • Check for continuity over their domains. All listed functions are continuous within their defined domains.
   • Analyze end behavior:
     • $f_1(x)$: increases without bound.
     • $f_2(x)$: increases rapidly as $x$ moves away from zero.
     • $f_3(x)$: increases but at a decreasing rate.
     • $f_4(x)$: increases dramatically as $x$ grows large.
     • $f_5(x)$: decreases without bound.
     • $f_6(x)$: increases linearly.

4. Assess Symmetry

   • Check for symmetry:
     • $f_1(x)$: Odd function (symmetric about origin).
     • $f_2(x)$: Even function (symmetric about $y$-axis).
     • $f_3(x)$: Neither even nor odd.
     • $f_4(x)$: Odd function.
     • $f_5(x)$: Odd function.
     • $f_6(x)$: Neither even nor odd.

5. Evaluate Special Points

   • Identify key features like intercepts and turning points:
     • $f_1(x)$: Intercept at (0,0).
     • $f_2(x)$: Intercept at (0,0).
     • $f_3(x)$: Intercept at (0,0).
     • $f_4(x)$: Intercept at (0,0).
     • $f_5(x)$: Intercept at (-2,0).
     • $f_6(x)$: Intercept at (-2,0).