Question Type 2: Determining if a specific relation is a function Exercises

Consider the parametrization of a circle given by $x=\cos t, \quad y=\sin t$ for $t \in [0, \pi/2]$.

Determine whether this parametrization defines $y$ as a function of $x$. Justify your answer using the relation between $x$ and $y$.

Given the relation $y = \sqrt{1 - x^2}$, determine if this defines $y$ as a function on $[-1,1]$ and justify your answer with reference to the definition of a function.

Solve the equation $x^2+y^2=1$ for $y$ and discuss whether the solutions define $y$ as a single-valued function on the domain $[-1,1]$.

Consider the mapping that assigns to each $x$ in $[-1,1]$ the two values $y = \pm\sqrt{1 - x^2}$. Does this mapping define a function? Explain.

Explain why there is no single real-valued continuous function $y=f(x)$ defined on $[-1,1]$ whose graph is the entire circle $x^2+y^2=1$.

Let $F(x,y) = x^2+y^2-1$. Use the implicit function theorem on $F(x,y)=0$ to determine where $y$ can be expressed as a differentiable function of $x$. Specify the regions in the plane.

Define $f(x)=\sqrt{1 - x^2}$ and $g(x)=-\sqrt{1 - x^2}$ on $[-1,1]$. Are $f$ and $g$ each functions on $[-1,1]$? Discuss whether either has an inverse that satisfies the circle equation. [4 marks]

Restrict the relation $x^2+y^2=1$ to the lower semicircle and write $y$ explicitly as a function of $x$. State the domain of this function.

Restrict the relation $x^2+y^2=1$ to the upper semicircle and write $y$ explicitly as a function of $x$. State the domain of this function.

Determine whether the implicit relation $x^2+y^2=1$ defines $y$ as a function of $x$ over the interval $[-1,1]$. Use the vertical line test to justify your answer.