QUESTION 1 1 Given xy = 2: 1.1 Sketch the graph of xy = 2. 1.2 Is this graph a function or a relation? 1.3 Is this graph continuous or discontinuous?

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QUESTION 1

1.1 Given $xy = 2$:

1.1.1 Sketch the graph of $xy = 2$.
The equation $xy = 2$ can be rewritten as $y = \frac{2}{x}$. This is a hyperbola.
The graph has two branches:

• One branch is in the first quadrant (where $x > 0$ and $y > 0$). It passes through points like $(1, 2)$ and $(2, 1)$.
• The other branch is in the third quadrant (where $x < 0$ and $y < 0$). It passes through points like $(-1, -2)$ and $(-2, -1)$. The graph has a vertical asymptote at $x=0$ (the y-axis) and a horizontal asymptote at $y=0$ (the x-axis). The branches approach these axes but never touch them.

1.1.2 Is this graph a function or a relation?
A function is a relation where each input ($x$-value) corresponds to exactly one output ($y$-value). For $y = \frac{2}{x}$, for every $x$ (except $x=0$), there is only one unique $y$ value. This graph passes the vertical line test.
Therefore, it is a function.

1.1.3 Is this graph continuous or discontinuous?
A graph is continuous if it can be drawn without lifting the pen. The graph of $y = \frac{2}{x}$ has a break at $x=0$ due to the vertical asymptote.
Therefore, it is discontinuous.