Functions and Relations (HSC SSCE Mathematics Advanced): Revision Notes

Worked Example: Determining if a Table Represents a Function

Question: This table represents a relation between $x$ and $y$. Determine if it represents a function.

Strategy:

A function requires each $x$-value to have exactly one $y$-value. We need to check the table for any repeated $x$-values with different $y$-values.

Solution:

Looking at the table, we can see that $x = 9$ appears twice:

• When $x = 9$, we have $y = -4$
• When $x = 9$, we also have $y = 11$

Since the $x$-value of $9$ corresponds to two different $y$-values ($-4$ and $11$), this is a one-to-many relationship.

Conclusion: This relation is NOT a function.

Worked Example: Testing a Reciprocal Function

Question: Determine whether this graph represents a function.

Strategy:

Apply the vertical line test. If no vertical line intersects the graph more than once, it's a function.

Solution:

The graph shows $y = \dfrac{1}{x}$ (a reciprocal function or hyperbola). When we imagine drawing vertical lines across the graph:

Each vertical line at any $x$-value intersects the curve at most once. For all $x$-values, each input produces exactly one $y$-value.

Conclusion: This graph represents both a relation and a function.

Note: All functions are relations, but not all relations are functions. Since each input has exactly one output, this is indeed a function.

Worked Example: Testing a Sideways Parabola

Question: Determine whether this graph represents a function.

Strategy:

Apply the vertical line test to check for multiple intersections.

Solution:

This graph appears to show a sideways parabola (like $x = y^2$ or $y = \pm\sqrt{x}$). When we draw a vertical line at $x = 1$:

The vertical line at $x = 1$ intersects the graph at two points: $(1, 1)$ and $(1, -1)$. This means the single $x$-value of $1$ corresponds to multiple $y$-values.

Conclusion: This graph represents a relation but NOT a function.

Reflection: If we restricted the graph to only $y \geq 0$ (the upper branch), it would then be a function, as each $x$-value would have only one corresponding $y$-value.

Worked Example: Testing a Linear Equation

Question: Determine whether the equation $y = 9x$ describes a relation and/or a function.

Strategy:

Graph the equation and apply the vertical line test.

Solution:

The equation $y = 9x$ is a linear equation that passes through the origin with a steep positive slope.

When we apply the vertical line test, every vertical line intersects the graph at exactly one point. This means each $x$-value produces precisely one $y$-value.

Conclusion: The equation describes both a relation and a function. Specifically, it's a one-to-one function since each input has a unique output.

Worked Example: Testing a Quadratic Equation

Question: Determine whether the equation $y = x^2 + 2$ describes a relation and/or a function.

Strategy:

Graph the equation and apply the vertical line test.

Solution:

The equation $y = x^2 + 2$ is a quadratic equation that forms a parabola opening upward with vertex at $(0, 2)$.

When we apply the vertical line test, every vertical line intersects the graph at exactly one point. Each $x$-value produces only one $y$-value.

Conclusion: The equation describes both a relation and a function. This is a many-to-one function because multiple inputs (like $x = -1$ and $x = 1$) can produce the same output ($y = 3$), but each individual input still produces only one output.

Reflection: While different $x$-values may produce the same $y$-value (many-to-one), this is still a function because the critical requirement is that each $x$-value has exactly one corresponding $y$-value.