Problem 3 - Algebra 2 and Trigonometry

Question

Let \(f=\\{(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)\\}\) and \(g=\\{(1,1),(2,4),(3,9),(4,16),(5,25)(6,36)\\}\) \(\begin{array}{ll}{\text { a. What is the domain of } \mathrm{f} \text { ? }} & {\text { b. What is the domain of } \mathrm{g} ?} \\ {\text { c. What is the domain of }(\mathrm{g}-\mathrm{f}) ?} & {\text { d. List the ordered pairs of }(\mathrm{g}-\mathrm{f}) \text { in set notation. }} \\ {\text { e. What is the domain of } \frac{\mathrm{g}}{\mathrm{f}} ?} & {\text { f. List the ordered pairs of } \frac{\mathrm{g}}{\mathrm{f}} \text { in set notation. }}\end{array}\)

Step-by-Step Solution

Answer

Domains: \(f: \{0, 1, 2, 3, 4, 5\}\); \(g: \{1, 2, 3, 4, 5, 6\}\); \(g-f: \{1, 2, 3, 4, 5\}\); \(\frac{g}{f}: \{1, 2, 3, 4\}\). Ordered pairs: \(g-f: \{(1, -3), (2, 1), (3, 7), (4, 15), (5, 25)\}\); \(\frac{g}{f}: \{(1, \frac{1}{4}), (2, \frac{4}{3}), (3, \frac{9}{2}), (4, 16)\}\).

Step 1: Domain of f

The domain of a function is the set of all first elements (or "x" values) from the ordered pairs. For the function \(f=\{(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)\}\), the domain is \(\{0, 1, 2, 3, 4, 5\}\).

Step 2: Domain of g

The domain of the function \(g=\{(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)\}\) is the set of all first elements (or "x" values) from the ordered pairs. This means the domain is \(\{1, 2, 3, 4, 5, 6\}\).

Step 3: Domain of (g-f)

The operation \((g-f)\) is defined for inputs that are present in the domains of both \(f\) and \(g\). Therefore, the domain of \((g-f)\) is the intersection of the domains of \(f\) and \(g\), which is \(\{1, 2, 3, 4, 5\}\).

Step 4: Ordered Pairs of (g-f)

Substitute each common domain element into both functions \(g\) and \(f\) and subtract. So, the ordered pairs for \((g-f)\) are: - For \(x=1\): \(g(1) - f(1) = 1 - 4 = -3\)- For \(x=2\): \(g(2) - f(2) = 4 - 3 = 1\)- For \(x=3\): \(g(3) - f(3) = 9 - 2 = 7\)- For \(x=4\): \(g(4) - f(4) = 16 - 1 = 15\)- For \(x=5\): \(g(5) - f(5) = 25 - 0 = 25\)Hence, \((g-f) = \{(1, -3), (2, 1), (3, 7), (4, 15), (5, 25)\}\).

Step 5: Domain of g/f

The domain of \(\frac{g}{f}\) is the intersection of the domains of \(g\) and \(f\), but excluding any elements where \(f(x) = 0\) (since division by zero is undefined). \(f(x) = 0\) at \(x=5\). Thus, the domain of \(\frac{g}{f}\) is \(\{1, 2, 3, 4\}\).

Step 6: Ordered Pairs of g/f

Evaluate the function \(\frac{g}{f}\) for each element of its domain:- For \(x=1\): \(\frac{g(1)}{f(1)} = \frac{1}{4}\)- For \(x=2\): \(\frac{g(2)}{f(2)} = \frac{4}{3}\)- For \(x=3\): \(\frac{g(3)}{f(3)} = \frac{9}{2}\)- For \(x=4\): \(\frac{g(4)}{f(4)} = \frac{16}{1}\)Therefore, the ordered pairs of \(\frac{g}{f}\) are \(\{(1, \frac{1}{4}), (2, \frac{4}{3}), (3, \frac{9}{2}), (4, 16)\}\).

Key Concepts

Set Notation

In mathematics, **set notation** is a {}``structured way to clearly define and represent a collection of items or numbers.`` It provides a standardized way to describe what elements belong to a set.

• A set is often written using curly braces \( \{ \} \), where the elements of the set are listed between these braces.
• For example, a set containing numbers from 1 to 5 would be written as \( \{ 1, 2, 3, 4, 5 \} \).
• Sets can also describe ordered pairs, where each pair consists of two elements. An example would be \(\{(1, 2), (3, 4)\}\).

Using set notation makes it straightforward to perform various operations in functions or other mathematical tasks. It facilitates the process of grouping particular values that meet a certain criterion.

Ordered Pairs

**Ordered pairs** are a fundamental concept in functions and coordinate geometry. '

• An ordered pair is written as \((x, y)\), where \(x\) is the first element and \(y\) is the second element.
• The order matters; \((x, y)\) is not the same as \((y, x)\).
• In the context of functions, the first element of an ordered pair typically represents an input value, while the second represents the corresponding output value.

Consider a function denoted by a series of ordered pairs, such as \(\{(0, 5), (1, 4), (2, 3)\}\). Here, each \(x\) value corresponds to a unique \(y\) value. Ordered pairs are crucial for plotting graphs and understanding how inputs relate to outputs in a function.

Intersection of Domains

An **intersection of domains** occurs when determining which input values are common to two or more functions. '

• The domain of a function is the complete set of possible values of the independent variable; think of it as all the possible \(x\) values that can go into a function.
• The intersection finds common \(x\) values that appear in each domain of the functions involved.
• For example, if function \(f\) has a domain of \(\{1, 2, 3, 4, 5\}\) and \(g\) has a domain of \(\{1, 2, 3, 4, 5, 6\}\), their intersection is \(\{1, 2, 3, 4, 5\}\).

Finding this intersection helps us understand which elements are viable for operations such as subtraction or division between two functions.

Function Operations

**Function operations** involve performing mathematical operations on functions such as addition, subtraction, multiplication, or division. '

• When two functions \(f\) and \(g\) are involved, one can create a new function such as \(f+g\), \(f-g\), \(f \cdot g\), or \(\frac{f}{g}\).
• The outcome of these operations is another function that combines the elements of the original functions.
• For example, the operation \(g-f\) takes the corresponding \(y\)-values from each function and subtracts them, generating a new set of ordered pairs. Similarly, \(\frac{g}{f}\) would divide the corresponding values from \(g\) by those from \(f\) when \(feq 0\).

These operations require careful consideration of domains to ensure that they are mathematically valid, particularly with division, where dividing by zero is undefined.