Section 3.2 Algebra Determining Functions Practice A

Diving deep into the fascinating realm of functions in algebra, Section 3.2 opens a door to understanding how to determine if a given relation truly qualifies as a function. This section is pivotal, acting as a cornerstone for more advanced algebraic concepts, from calculus to linear algebra. Mastering the principles outlined in Section 3.2 allows you to confidently identify functions, analyze their properties, and predict their behavior. This practice will provide a thorough understanding of how these principles work in reality.

Understanding the Core Concepts

Before we dive into practice problems, let's solidify our understanding of what defines a function and the tools we use to identify them.

What is a Function?

At its heart, a function is a special type of relationship between two sets, often called the domain and the range. Think of the domain as the set of possible inputs and the range as the set of possible outputs. The defining characteristic of a function is that each input in the domain is associated with exactly one output in the range. This "one-to-one" or "many-to-one" relationship (but never "one-to-many") is what distinguishes a function from a general relation.

Key Terminology:

• Domain: The set of all possible input values (often represented by x).
• Range: The set of all possible output values (often represented by y).
• Relation: Any set of ordered pairs.
• Function: A relation where each input has only one output.
• Independent Variable: The input variable (typically x).
• Dependent Variable: The output variable (typically y), its value depends on the input.

Representing Functions:

Functions can be represented in several ways, each offering a unique perspective:

• Equations: A mathematical statement defining the relationship between x and y (e.g., y = 2x + 1).
• Graphs: A visual representation plotting ordered pairs (x, y) on a coordinate plane.
• Tables: A tabular representation listing corresponding x and y values.
• Mappings: A diagram illustrating the association between elements of the domain and range.
• Set of Ordered Pairs: Listing the (x, y) pairs explicitly.

The Vertical Line Test:

A powerful visual tool for determining if a graph represents a function is the Vertical Line Test. If any vertical line drawn on the graph intersects the graph at more than one point, then the graph does not represent a function. This is because a single x-value would be associated with multiple y-values, violating the fundamental definition of a function.

Practice Problems and Solutions

Let's work through a series of practice problems covering various representations of relations and apply our knowledge to determine if they are functions.

Problem 1: Set of Ordered Pairs

Determine if the following set of ordered pairs represents a function:

A = { (1, 2), (2, 4), (3, 6), (4, 8), (1, 3) }

Solution:

To determine if this set represents a function, we need to check if any x-value is paired with more than one y-value. We observe that the x-value of 1 is paired with both 2 and 3. Therefore, this set of ordered pairs does not represent a function.

Problem 2: Table of Values

Determine if the following table of values represents a function:

Solution:

In this table, each x-value has a unique y-value. Even though the y-value of 4 appears twice, it's associated with different x-values (-2 and 2). Therefore, this table does represent a function.

Problem 3: Equation

Determine if the following equation represents a function:

y = x² + 3

Solution:

For any given value of x, the equation y = x² + 3 will produce only one value of y. For example, if x = 2, then y = 2² + 3 = 7. There's no ambiguity; each x maps to a single y. Therefore, this equation does represent a function.

Problem 4: Equation (Circle)

Determine if the following equation represents a function:

x² + y² = 9

Solution:

This equation represents a circle centered at the origin with a radius of 3. Consider the x-value of 0. Substituting x = 0 into the equation, we get:

0² + y² = 9 y² = 9 y = ±3

So, when x = 0, y can be either 3 or -3. This means the point (0, 3) and (0, -3) both lie on the graph. This violates the definition of a function, because one x-value is associated with two different y-values. Alternatively, if you were to graph the equation, you would clearly see that it fails the vertical line test. Therefore, this equation does not represent a function.

Problem 5: Graph

Determine if the following graph represents a function (imagine a parabola opening to the right):

[Imagine a parabola graphed on a coordinate plane, but opening sideways (to the right) instead of up or down.]

Solution:

A parabola opening to the right fails the vertical line test. For example, a vertical line drawn at x = 2 would intersect the parabola at two points. Therefore, this graph does not represent a function.

Problem 6: Piecewise Function

Determine if the following piecewise function represents a function:

f(x) = { x + 1, if x < 0 x², if x ≥ 0 }

Solution:

A piecewise function defines different rules for different intervals of the domain. To determine if it's a function, we need to ensure that there's no overlap in the y-values at the boundaries of the intervals. In this case, the boundary is at x = 0.

• For x < 0, f(x) = x + 1. As x approaches 0 from the left, f(x) approaches 1.
• For x ≥ 0, f(x) = x². At x = 0, f(x) = 0² = 0.

Since the function is defined at x=0 as f(x) = x² = 0, and x+1 is defined only for x<0, we can conclude that this piecewise function does represent a function, because even though it approaches 1 from the left, f(0) is strictly defined as 0.

Problem 7: Mapping Diagram

Determine if the following mapping diagram represents a function:

[Imagine a mapping diagram with the domain {A, B, C} and the range {1, 2, 3}. A maps to 1, B maps to 2, and C maps to both 2 and 3.]

Solution:

In this mapping diagram, the element C in the domain maps to two different elements (2 and 3) in the range. This violates the definition of a function. Therefore, this mapping diagram does not represent a function.

Problem 8: Absolute Value Function

Determine if the following equation represents a function:

y = |x|

Solution:

The absolute value function, y = |x|, assigns a unique non-negative value to each input x. For example, if x = -3, y = |-3| = 3. If x = 3, y = |3| = 3. Although different x-values can map to the same y-value (as in this case), each x-value maps to only one y-value. Therefore, this equation does represent a function.

Problem 9: A More Complex Equation

Determine if the following equation represents a function:

y³ = x

Solution:

For every value of x, there is only one real number y that satisfies the equation y³ = x. For example, if x = 8, then y³ = 8, and the only real solution is y = 2. We can also rewrite this equation as y = ∛x, which clearly shows that for each x there is only one cube root y. Therefore, this equation does represent a function.

Problem 10: Rational Function

Determine if the following equation represents a function:

y = 1/x

Solution:

For every value of x except x = 0, the equation y = 1/x produces exactly one value of y. When x = 0, the expression is undefined. However, the absence of a value at x = 0 doesn't violate the definition of a function; it simply means 0 is not in the domain. As long as each x in the domain is associated with only one y-value, the relation is a function. Therefore, this equation does represent a function.

Common Pitfalls and Misconceptions

Understanding common mistakes can significantly improve your ability to identify functions correctly.

• Confusing "One-to-One" with "Function": While a one-to-one function is a function, not all functions are one-to-one. A function can be "many-to-one" (multiple x-values mapping to the same y-value) and still be a function. The crucial point is that one x-value cannot map to multiple y-values.
• Assuming All Equations are Functions: Many equations are not functions. The equation of a circle is a classic example. Always test the equation (or graph it) to verify if it meets the function criteria.
• Ignoring the Domain: The domain plays a crucial role. An equation might be a function for a restricted domain, but not for all real numbers. For instance, y = √x is a function if we consider only non-negative values of x.
• Misapplying the Vertical Line Test: The vertical line test only applies to graphs. Do not attempt to use it on equations or tables of values directly.
• Overlooking Piecewise Functions: Carefully examine piecewise functions at the boundaries of their intervals to ensure there's no ambiguity in the y-values.

The Importance of Determining Functions

The ability to confidently determine whether a relation is a function is fundamental to further study in mathematics. Functions are the building blocks of more advanced concepts like:

• Calculus: Derivatives and integrals operate on functions.
• Linear Algebra: Linear transformations are functions between vector spaces.
• Differential Equations: Solutions to differential equations are functions.
• Modeling: Real-world phenomena are often modeled using functions.

Without a solid understanding of functions, you will struggle to grasp these more advanced topics.

Advanced Considerations

While the basics are crucial, exploring some advanced considerations can deepen your understanding.

• Injective, Surjective, and Bijective Functions: These classifications further refine the types of functions.
  • Injective (One-to-One): Each element of the range is associated with at most one element of the domain. (Passes both vertical and horizontal line tests).
  • Surjective (Onto): Every element of the range is associated with at least one element of the domain (the range is equal to the codomain).
  • Bijective: Both injective and surjective.
• Inverse Functions: Only bijective functions have inverse functions. The inverse function "undoes" the original function. If f(x) = y, then f^-1(y) = x.
• Function Composition: Combining two functions to create a new function. If f(x) and g(x) are functions, then the composition f(g(x)) is also a function (provided the range of g(x) is a subset of the domain of f(x)).
• Implicit Functions: Functions defined implicitly by an equation (e.g., x² + y² = 9 can define y as a function of x over restricted intervals).

Further Practice

To truly master the concepts in Section 3.2, consistent practice is essential. Here are some additional exercises:

1. Determine if the following sets of ordered pairs represent functions:

   • {(2, 5), (3, 7), (4, 9), (2, 6)}
   • {(-1, 0), (0, 1), (1, 2), (2, 3)}

2. Determine if the following tables of values represent functions:

   x	y
   1	2
   2	2
   3	2
   4	2

   x	y
   -3	9
   -2	4
   -1	1
   -3	0

3. Determine if the following equations represent functions:

   • y = x³ - 2x + 1
   • x = y²
   • y = √(x + 4)
   • x² + (y - 2)² = 16

4. Sketch the graphs of the following equations and use the vertical line test to determine if they represent functions:

   • y = -x + 3
   • y = |x - 1|
   • x² + y² = 25
   • x = y³

Conclusion

Section 3.2 of algebra, focusing on determining functions, is a foundational element in your mathematical journey. By understanding the definition of a function, recognizing its various representations, and practicing with a diverse range of examples, you can build a strong foundation for more advanced mathematical concepts. Remember to be mindful of common pitfalls and misconceptions, and continue to explore the nuances of functions to deepen your understanding. The more you practice, the more intuitive the process will become. Good luck!