Solutions - 0265-02

(a)

Observe that $x=t$ and $y=t$ implies $y=x$.

Therefore, all points on the curve satisfy $y=x$ and we set $f(x)=x$.

Since $x=t$ and $t$ covers the entire real line $x\in (-\infty ,\infty )$, the parametric curve is the entire line $y=x$.

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(b)

Observe that $x=e^t$ and $y=e^t$ implies $y=x$. Again, all points on the curve satisfy $y=x$ and so $f(x)=x$.

However, this time $t\in (-\infty ,\infty )$ implies $x>0$, and the entire range of $x>0$ is possible (set $t=\ln x$) to find an inverse.

So the image of this parametric curve is $y=x$ for $x>0$, and the origin is omitted.