Question

describe the listed aspects of the function. be sure to label and describe.
function:
( g(x) = x^3 + 2x^2 - x + 4 )

• graph
• domain and range
• continuity
• symmetry
• extrema
• increasing/decreasing intervals
• end behavior

(handwritten: domain: ( x = (-infty, infty) ); range: ( y = (-infty, infty) ); contionous (sic); rotation symmetry; inc: ( x = -infty, 1.53 cup 0.27, infty ); dec: ( x = -1.53, 0.27 ); ( \begin{cases} lim_{x \to -infty} f(x) = -infty \\ lim_{x \to infty} f(x) = infty end{cases} ); graph sketch with relative max/min, axes)

Answer:

1. Graph

The function \( g(x) = x^3 + 2x^2 - x + 4 \) is a cubic (degree 3) polynomial. The leading coefficient is positive (\( 1 \)), so the graph has the characteristic "S - shaped" curve of a cubic: it falls to the left (as \( x \to -\infty \)) and rises to the right (as \( x \to \infty \)). It has a relative maximum and a relative minimum (visible in the sketch), with the curve passing through these extrema and being continuous everywhere.

2. Domain and Range

• Domain: For polynomial functions (like cubics), there are no restrictions on \( x \) (no denominators, square roots, etc. that limit \( x \)-values). So the domain is all real numbers, written as \( (-\infty, \infty) \) (or \( \mathbb{R} \)).
• Range: Cubic functions with odd degrees (degree 3) have a range of all real numbers, since they extend infinitely in both the positive and negative \( y \)-directions. Thus, the range is \( (-\infty, \infty) \) (or \( \mathbb{R} \)).

3. Continuity

Polynomial functions are continuous everywhere on their domain (which is all real numbers for cubics). This means there are no breaks, holes, or asymptotes in the graph of \( g(x) \); it can be drawn without lifting the pencil from the paper.

4. Symmetry

• Test for Symmetry:
• Even Function (Symmetric about \( y \)-axis): Check if \( g(-x) = g(x) \).

\( g(-x) = (-x)^3 + 2(-x)^2 - (-x) + 4 = -x^3 + 2x^2 + x + 4 \), which is not equal to \( g(x) = x^3 + 2x^2 - x + 4 \).

• Odd Function (Symmetric about origin): Check if \( g(-x) = -g(x) \).

\( -g(x) = -x^3 - 2x^2 + x - 4 \), which is not equal to \( g(-x) \).

• The sketch suggests "rotation symmetry" (likely about the origin or a point), but algebraically, it is not an even or odd function. However, cubic functions can have point - symmetry (rotation symmetry of \( 180^\circ \) about a point). To find the point of symmetry, for a cubic \( ax^3+bx^2 + cx + d \), the point of symmetry has \( x \)-coordinate \( x=-\frac{b}{3a} \). For \( g(x)=x^3 + 2x^2 - x + 4 \), \( a = 1 \), \( b = 2 \), so \( x=-\frac{2}{3(1)}=-\frac{2}{3} \). Substitute \( x = -\frac{2}{3} \) into \( g(x) \) to find the \( y \)-coordinate of the point of symmetry: \( g(-\frac{2}{3})=(-\frac{2}{3})^3+2(-\frac{2}{3})^2-(-\frac{2}{3}) + 4=-\frac{8}{27}+\frac{8}{9}+\frac{2}{3}+4=\frac{- 8 + 24+18 + 108}{27}=\frac{142}{27}\approx5.26 \). So the graph is symmetric about the point \( (-\frac{2}{3},\frac{142}{27}) \) (rotation symmetry of \( 180^\circ \) about this point).

5. Extrema

To find extrema, we take the derivative: \( g'(x)=3x^2 + 4x - 1 \). Set \( g'(x)=0 \) and solve for \( x \) using the quadratic formula \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 4 \), \( c=-1 \).
\( x=\frac{-4\pm\sqrt{16+12}}{6}=\frac{-4\pm\sqrt{28}}{6}=\frac{-4\pm2\sqrt{7}}{6}=\frac{-2\pm\sqrt{7}}{3} \approx\frac{-2\pm2.6458}{3} \).
So \( x_1\approx\frac{-2 + 2.6458}{3}\approx0.215 \) and \( x_2\approx\frac{-2-2.6458}{3}\approx - 1.548 \).

• Relative Maximum: At \( x\approx - 1.548 \), substitute into \( g(x) \): \( g(-1.548)\approx(-1.548)^3+2(-1.548)^2-(-1.548)+4\approx - 3.75+4.79 + 1.55+4\approx6.59 \).
• Relative Minimum: At \( x\approx0.215 \), substitute into \( g(x) \): \( g(0.215)\approx(0.215)^3+2(0.215)^2-(0.215)+4\approx0.01 + 0.09 - 0.22+4\approx3.88 \).

6. Increasing/Decreasing Intervals

Using the critical points \( x=\frac{-2-\sqrt{7}}{3}\approx - 1.548 \) and \( x=\frac{-2+\sqrt{7}}{3}\approx0.215 \), we test the sign of \( g'(x) \) in the intervals:

• Interval \( (-\infty,\frac{-2 - \sqrt{7}}{3}) \): Pick \( x=-2 \), \( g'(-2)=3(4)+4(-2)-1 = 12 - 8 - 1 = 3>0 \), so \( g(x) \) is increasing.
• Interval \( (\frac{-2 - \sqrt{7}}{3},\frac{-2+\sqrt{7}}{3}) \): Pick \( x = 0 \), \( g'(0)=-1<0 \), so \( g(x) \) is decreasing.
• Interval \( (\frac{-2+\sqrt{7}}{3},\infty) \): Pick \( x = 1 \), \( g'(1)=3 + 4 - 1 = 6>0 \), so \( g(x) \) is increasing.

Thus, \( g(x) \) is increasing on \( (-\infty,\frac{-2 - \sqrt{7}}{3})\cup(\frac{-2+\sqrt{7}}{3},\infty) \) and decreasing on \( (\frac{-2 - \sqrt{7}}{3},\frac{-2+\sqrt{7}}{3}) \).

7. End Behavior

For a polynomial \( a_nx^n+\cdots+a_0 \), the end behavior is determined by the leading term \( a_nx^n \). For \( g(x)=x^3+2x^2 - x + 4 \), the leading term is \( x^3 \) ( \( n = 3 \), odd, \( a_n=1>0 \) ).

• As \( x\to-\infty \), \( x^3\to-\infty \), so \( \lim_{x\to-\infty}g(x)=-\infty \).
• As \( x\to\infty \), \( x^3\to\infty \), so \( \lim_{x\to\infty}g(x)=\infty \).

Final Answer (Summary)

• Graph: Cubic curve, falls left, rises right, with one relative max and one relative min.
• Domain: \( (-\infty, \infty) \); Range: \( (-\infty, \infty) \)
• Continuity: Continuous everywhere on \( \mathbb{R} \).
• Symmetry: Rotation symmetry about \( (-\frac{2}{3},\frac{142}{27}) \) (not even/odd).
• Extrema: Relative max at \( x\approx - 1.55,y\approx6.59 \); relative min at \( x\approx0.22,y\approx3.88 \).
• Increasing/Decreasing: Increasing on \( (-\infty,-1.55)\cup(0.22,\infty) \), decreasing on \( (-1.55,0.22) \).
• End Behavior: \( \lim_{x\to-\infty}g(x)=-\infty \), \( \lim_{x\to\infty}g(x)=\infty \).