Given the function [math]f\left(x\right)=\frac{x^3+3x^2}{x^2+6x+9}[/math].[br][list=1][*]Plot the graph of [i]f(x)[/i].[/*][*]Determine the roots and maximum domain [i]D[/i] of [i]f(x)[/i].[/*][*]Identify the behaviour of [i]f(x)[/i] at the edges of [i]D[/i].[/*][/list]

[table][tr][td]1.[/td][td]Define the function [i]f(x)[/i] by entering [math]f\left(x\right)=\frac{x^3+3x^2}{x^2+6x+9}[/math] into the [i]Input Bar [/i]and pressing [i]Enter[/i]. [/td][/tr][tr][td][/td][td][b]Note:[/b] [i]GeoGebra CAS Calculator[/i] automatically simplifies the equation of [i]f(x)[/i].[/td][/tr][tr][td]2.[/td][td]Use the command [math]Root(f)[/math] to determine the roots of [i]f(x)[/i].[br][/td][/tr][tr][td][/td][td][b]Note: [/b]You can also use the command [math]Solve(f=0)[/math] to calculate the roots of [i]f(x)[/i].[/td][/tr][tr][td]3.[/td][td]To determine the maximum domain of f(x), calculate the roots of the denominator of f(x) by entering the command [math]Root(Denominator(f))[/math].[br][/td][/tr][tr][td][/td][td][b]Note:[/b] You can also use the command [math]Solve(Denominator(f)=0)[/math].[/td][/tr][tr][td]4.[/td][td]As the solution is [i]-3[/i] we get the maximum domain [math]D=\mathbb{R}\backslash\left\{-3\right\}[/math].[/td][/tr][/table]

[table][tr][td]5.[/td][td]Use the [i]Limit[/i] command to identify the behaviour of [i]f(x)[/i] at the edges of [i]D[/i]. [/td][/tr][tr][td][/td][td]Enter the command [math]Limit\left(f,-\infty\right)[/math] into the [i]Input Bar[/i] to identify the behaviour on the left edge.[/td][/tr][tr][td][/td][td]Enter the command [math]Limit\left(f,\infty\right)[/math] into the [i]Input Bar[/i] to identify the behaviour on the right edge.[/td][/tr][tr][td]6.[/td][td]Use the commands [i]LimitBelow[/i] and [i]LimitAbove[/i] to identify the behaviour of [i]f(x)[/i] around [i]-3[/i]. [/td][/tr][tr][td][/td][td]Enter the command [math]LimitBelow\left(f,-3\right)[/math] into the [i]Input Bar[/i].[/td][/tr][tr][td][/td][td]Enter the command [math]LimitAbove\left(f,-3\right)[/math] into the [i]Input Bar[/i].[/td][/tr][tr][td][/td][td][br][/td][/tr][/table]