Chapter 1.5, Problem 37E

Solution

To identify: The basic function who parametric equation is given.

The basic function represented by the given parametric equation is $f\left(x\right)=x$ , and its domain is $\left(-\infty ,\infty \right)$ .

Given information: The parametric equation of the function is $x=t^3,y=\sqrt{t^6},-\infty <t<\infty .$

Explanation:

The basic function from the parametric curve is obtained by eliminating the parameter from the given equations.

The given parametric equations are; $x=t^3,y=\sqrt{t^6},-\infty <t<\infty .$ , then the expression for $y/x$ is

  $\begin{array}{c}\frac{y}{x}=\frac{\sqrt{t^6}}{t^3}\\ \frac{y}{x}=\frac{{\left(t^6\right)}^{\frac{1}{2}}}{t^3}\\ \frac{y}{x}=\frac{{\left(t^{\cancel{6}3}\right)}^{\frac{1}{\cancel{2}}}}{t^3}\\ \frac{y}{x}=\frac{t^3}{t^3}\\ \frac{y}{x}=1\Rightarrow y=x\end{array}$

Write $y=f\left(x\right)$ , therefore, $f\left(x\right)=x$ is the basic function represented by parametric equations. Since, the function $f\left(x\right)=x$ is defined for all $x\in \left(-\infty ,\infty \right)$ . Therefore, the domain of the function $f\left(x\right)=x$ is $\left(-\infty ,\infty \right)$ .