Question

What is the image of a function ffrom $\mathbf{ℝ}$ to $\mathbf{ℝ}$given by

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{t}}^{\mathbf{3}}\mathbf{+}\mathit{a}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathit{b}\mathit{t}\mathbf{+}\mathit{c}$,

where a,b,c are arbitrary scalars?

Short answer

The image is (f) $=\left\{f\left(t\right)=t^3+a{t}^2+bt+c:tinT\right\}$ ,image is all of$\mathrm{ℝ}$ .

Step-by-step solution

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$$\begin{gathered} image\left(f\right)=\left\{f\left(x\right):inX\right\} \\ =\left\{binY:b=f\left(x\right),\mathrm{for}\mathrm{some}x\mathrm{in}X\right\} \end{gathered}$$

The Kernel of the transformation results in $\left\{f\left(t\right)=t^3+a{t}^2+bt+c:tinT\right\}$, as the reflection is its own inverse.

As the inverse of the reflection exists thus, the image will be $\mathrm{ℝ}$. The kernel is of dimension $image\left(f\right)=\left\{f\left(t\right)=t^3+a{t}^2+bt=c:tinT\right\}$ so, the image will be of $\mathrm{ℝ}$.

Final answer.

The image is $image\left(f\right)=\left\{f\left(t\right)=t^3+a{t}^2+bt+c:tinT\right\}$, image is all of $\mathrm{ℝ}$.