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

Determine whether the relation defined by $y = \frac{1}{x}\quad (x\ne0)$ is a function. Provide reasoning.

Consider the set of ordered pairs $R = \{(1,2),(1,3),(2,4)\}$. Determine whether $R$ represents a function from its first component to its second component.

Determine whether the implicit relation $x^2 + y^2 = 1$ defines $y$ as a function of $x$. Justify your answer.

Analyze the implicit relation $x^2 - y = 0$ and determine whether it defines $y$ as a function of $x$.

Given the mapping $M = \{(-1,1),(0,0),(1,1),(2,4)\},$ determine whether $M$ defines a function from its first component to its second component. Justify your conclusion.

Given the implicit equation $y^2 = x + 3$, determine whether $y$ is a function of $x$. Explain your reasoning.

Is the relation $y = \pm\sqrt{4 - x}$ a function of $x$? Explain your answer.

Determine if the relation defined by $x = (y - 2)^2$ expresses $y$ as a function of $x$. Provide justification.

Analyze the relation given by $|y| = x - 1$. Determine if $y$ is a function of $x$ and explain why.

Let $f(x) = x^2 + 1$ with domain $x\ge0$. Determine whether the inverse relation $f^{-1}$ is a function. Justify your answer.

Consider the parametric equations $x = t^2 - 4\,,\quad y = 2t + 1\,.$ Determine whether the resulting relation defines $y$ as a function of $x$. Explain.

Consider the parametric curve $x = \sin t\,,\quad y = \cos t\quad(t\in\mathbb{R}).$ Determine whether this relation defines $y$ as a function of $x$. Justify.