Question

Find the domain and sketch the graph of the functions\({\bf{f}}\left( {\bf{t}} \right){\bf{ = 2t + }}{{\bf{t}}^{\bf{2}}}\).

Short answer

The domain of the function\({\bf{f}}\left( {\bf{t}} \right){\bf{ = 2t + }}{{\bf{t}}^{\bf{2}}}\)is\(\left( { - \infty ,\infty } \right)\)and the graph of the function is given in figure (1).

Step-by-step solution

Determine the domain of the function

The domain of a function is the set of all possible values for which a function is defined.

The given function\(f\left( t \right) = 2t + {t^2}\)holds true for all the real numbers. Therefore given function is defined for\(t \in \left( { - \infty ,\infty } \right)\).

Therefore, the domain of the function \(f\left( t \right) = 2t + {t^2}\)is\(\left( { - \infty ,\infty } \right)\).

Sketch the graph of the function

The intersection points of the graph on t axis are calculated by substituting\(f\left( t \right) = 0\)in the function.

\(\begin{aligned}0 &= 2t + {t^2}\\t\left( {t + 2} \right) &= 0\\t &= 0, - 2\end{aligned}\)

The graph of the function is given below as:

Figure (1)

Therefore, the graph of the function is given in figure (1).