Guided Study for Prealgebra, Beginning Algebra, & Intermediate Algebra (with Relevant Images)

Q1. Of the following relations, circle those that represent functions:

• (a) $x = y^2$

• (b) $y = x^2$

• (c) \{(1,1), (1,2)\}

• (d) \{(1,1), (2,1)\}

• (e) $x = |y|$

Background

Topic: Functions and Relations

This question tests your understanding of what makes a relation a function. A function is a relation in which each input (usually $x$) is paired with exactly one output (usually $y$).

Key Terms and Concepts:

• Relation: A set of ordered pairs.

• Function: A relation where each input has exactly one output.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Step-by-Step Guidance

1. For each option, determine if any input value (for $x$ or $y$) is paired with more than one output.

2. For equations, try solving for $y$ in terms of $x$ (or vice versa) and see if each $x$ gives only one $y$.

3. For sets of ordered pairs, check if any $x$ value is repeated with a different $y$ value.

4. For graphs, imagine the vertical line test: would a vertical line ever cross the graph more than once?

5. Decide which options meet the definition of a function, but do not circle them yet—think through each one carefully.

Try solving on your own before revealing the answer!

Final Answer:

The relations that represent functions are:

• (b) $y = x^2$ (each $x$ gives only one $y$)

• (d) \{(1,1), (2,1)\} (no $x$ value repeats with a different $y$)

Explanation: In (a) and (e), a single $x$ value can correspond to more than one $y$ value. In (c), the $x$ value 1 is paired with both 1 and 2, so it is not a function.