Find Each Of The Following Functions And State Their Domains

Understanding functions and their domains is fundamental to mastering mathematics. Each function operates under specific rules, and its domain defines the set of input values for which the function is valid. This article will delve into various functions, illustrating how to find them and determine their corresponding domains.

Identifying and Defining Functions

Before we dive into specific examples, let's establish a clear understanding of what a function is. In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Key Components of a Function:

• Input (x): The value that you feed into the function. This is often called the independent variable.
• Function Rule (f(x)): The operation or set of operations performed on the input. This rule determines how the input is transformed into the output.
• Output (y or f(x)): The value that results from applying the function rule to the input. This is often called the dependent variable.
• Domain: The set of all possible input values (x) for which the function is defined.
• Range: The set of all possible output values (y) that the function can produce.

Types of Functions:

Functions come in various forms, each with its unique characteristics and domain considerations:

• Linear Functions: f(x) = mx + b, where m and b are constants.
• Quadratic Functions: f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.
• Polynomial Functions: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aᵢ are constants and n is a non-negative integer.
• Rational Functions: f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions.
• Radical Functions: f(x) = √x or f(x) = ∛x, involving roots of variables.
• Exponential Functions: f(x) = aˣ, where a is a constant and a > 0, a ≠ 1.
• Logarithmic Functions: f(x) = logₐ(x), where a is a constant and a > 0, a ≠ 1.
• Trigonometric Functions: f(x) = sin(x), f(x) = cos(x), f(x) = tan(x), and their reciprocals.

Determining the Domain of a Function

The domain of a function is a critical aspect because it dictates the valid input values. To find the domain, you must identify any restrictions on the input. Common restrictions include:

• Division by Zero: The denominator of a rational function cannot be zero.
• Square Roots of Negative Numbers: The radicand (the value under the root) of a square root function cannot be negative. In general, for even-indexed radicals, the radicand must be non-negative.
• Logarithms of Non-Positive Numbers: The argument of a logarithm function must be positive.

Let's explore how to find the domain for various types of functions with illustrative examples.

1. Polynomial Functions

Polynomial functions are defined for all real numbers. Therefore, their domain is always all real numbers, denoted as (-∞, ∞) or ℝ.

Example:

• f(x) = 3x² + 2x - 1

  Domain: (-∞, ∞)

2. Rational Functions

Rational functions have the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials. The domain consists of all real numbers except those that make the denominator q(x) equal to zero.

Example 1:

• f(x) = 1 / (x - 2)

  To find the domain, set the denominator equal to zero:

  x - 2 = 0 x = 2

  Therefore, x cannot be 2.

  Domain: (-∞, 2) ∪ (2, ∞)

Example 2:

• f(x) = (x + 1) / (x² - 4)

  Factor the denominator:

  x² - 4 = (x - 2)(x + 2)

  Set each factor equal to zero:

  x - 2 = 0 => x = 2 x + 2 = 0 => x = -2

  Therefore, x cannot be 2 or -2.

  Domain: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)

3. Radical Functions

Radical functions involve roots, such as square roots, cube roots, etc. The domain depends on the index of the root.

• Even-Indexed Roots (e.g., square root, fourth root): The radicand (the expression under the root) must be greater than or equal to zero.

• Odd-Indexed Roots (e.g., cube root, fifth root): The radicand can be any real number.

Example 1: Square Root

• f(x) = √(x - 3)

  The radicand must be non-negative:

  x - 3 ≥ 0 x ≥ 3

  Domain: [3, ∞)

Example 2: Cube Root

• f(x) = ∛(2x + 1)

  Since it's a cube root, the radicand can be any real number.

  Domain: (-∞, ∞)

4. Exponential Functions

Exponential functions have the form f(x) = aˣ, where a is a constant and a > 0, a ≠ 1. Exponential functions are defined for all real numbers.

Example:

• f(x) = 2ˣ

  Domain: (-∞, ∞)

5. Logarithmic Functions

Logarithmic functions have the form f(x) = logₐ(x), where a is a constant and a > 0, a ≠ 1. The argument of the logarithm must be positive.

Example 1:

• f(x) = ln(x + 2) (ln denotes the natural logarithm, base e)

  The argument must be positive:

  x + 2 > 0 x > -2

  Domain: (-2, ∞)

Example 2:

• f(x) = log₁₀(4 - x)

  The argument must be positive:

  4 - x > 0 4 > x x < 4

  Domain: (-∞, 4)

6. Trigonometric Functions

Trigonometric functions have varying domains depending on the specific function.

• Sine and Cosine: f(x) = sin(x) and f(x) = cos(x) are defined for all real numbers.

  Domain: (-∞, ∞)

• Tangent: f(x) = tan(x) = sin(x) / cos(x) is undefined where cos(x) = 0. This occurs at x = (π/2) + nπ, where n is an integer.

  Domain: All real numbers except x = (π/2) + nπ, where n is an integer.

• Cotangent: f(x) = cot(x) = cos(x) / sin(x) is undefined where sin(x) = 0. This occurs at x = nπ, where n is an integer.

  Domain: All real numbers except x = nπ, where n is an integer.

• Secant: f(x) = sec(x) = 1 / cos(x) is undefined where cos(x) = 0. This occurs at x = (π/2) + nπ, where n is an integer.

  Domain: All real numbers except x = (π/2) + nπ, where n is an integer.

• Cosecant: f(x) = csc(x) = 1 / sin(x) is undefined where sin(x) = 0. This occurs at x = nπ, where n is an integer.

  Domain: All real numbers except x = nπ, where n is an integer.

Finding Functions from Given Information

Sometimes, you might be given information about a function and need to determine the function's equation. This often involves using given points, slopes, or other characteristics to construct the function.

1. Linear Functions

Given two points (x₁, y₁) and (x₂, y₂), you can find the equation of the linear function using the slope-intercept form y = mx + b.

1. Calculate the slope (m): m = (y₂ - y₁) / (x₂ - x₁)
2. Use one of the points and the slope to find the y-intercept (b): y₁ = mx₁ + b => b = y₁ - mx₁
3. Write the equation: f(x) = mx + b

Example:

Given the points (1, 3) and (2, 5), find the linear function.

1. m = (5 - 3) / (2 - 1) = 2 / 1 = 2

2. Using the point (1, 3): 3 = 2(1) + b => b = 3 - 2 = 1

3. f(x) = 2x + 1

   Domain: (-∞, ∞)

2. Quadratic Functions

Finding a quadratic function can be more complex. If you're given the vertex (h, k) and another point (x, y), you can use the vertex form f(x) = a(x - h)² + k.

1. Substitute the vertex (h, k) into the vertex form: f(x) = a(x - h)² + k
2. Substitute the other point (x, y) into the equation and solve for a: y = a(x - h)² + k
3. Write the equation: f(x) = a(x - h)² + k

Example:

Given the vertex (2, 1) and the point (3, 2), find the quadratic function.

1. f(x) = a(x - 2)² + 1

2. Substitute (3, 2): 2 = a(3 - 2)² + 1 => 2 = a(1)² + 1 => a = 1

3. f(x) = (x - 2)² + 1 = x² - 4x + 4 + 1 = x² - 4x + 5

   Domain: (-∞, ∞)

3. Exponential Functions

If you have two points (x₁, y₁) and (x₂, y₂) on an exponential function of the form f(x) = a ⋅ bˣ, you can find the values of a and b.

1. Set up two equations using the given points:
   • y₁ = a ⋅ bˣ¹
   • y₂ = a ⋅ bˣ²
2. Divide the second equation by the first equation to eliminate a: (y₂ / y₁) = (bˣ² / bˣ¹)
3. Solve for b: b = (y₂ / y₁)^(1 / (x₂ - x₁))
4. Substitute the value of b into one of the original equations and solve for a: y₁ = a ⋅ bˣ¹ => a = y₁ / bˣ¹
5. Write the equation: f(x) = a ⋅ bˣ

Example:

Given the points (0, 2) and (1, 6), find the exponential function.

1. 2 = a ⋅ b⁰ 6 = a ⋅ b¹

2. 6 / 2 = (a ⋅ b¹) / (a ⋅ b⁰) => 3 = b

3. b = 3

4. 2 = a ⋅ 3⁰ => 2 = a ⋅ 1 => a = 2

5. f(x) = 2 ⋅ 3ˣ

   Domain: (-∞, ∞)

Combining Functions and Their Domains

Functions can be combined through operations such as addition, subtraction, multiplication, and division. When combining functions, the domain of the resulting function must take into account the domains of the individual functions and any new restrictions introduced by the combination.

1. Addition and Subtraction: (f + g)(x) = f(x) + g(x) and (f - g)(x) = f(x) - g(x)

The domain of (f + g)(x) and (f - g)(x) is the intersection of the domains of f(x) and g(x).

Example:

• f(x) = √(x - 1) Domain: [1, ∞)

• g(x) = x + 2 Domain: (-∞, ∞)

  (f + g)(x) = √(x - 1) + x + 2

  Domain: [1, ∞) ∩ (-∞, ∞) = [1, ∞)

2. Multiplication: (f ⋅ g)(x) = f(x) ⋅ g(x)

The domain of (f ⋅ g)(x) is the intersection of the domains of f(x) and g(x).

Example:

• f(x) = x / (x + 1) Domain: (-∞, -1) ∪ (-1, ∞)

• g(x) = x - 1 Domain: (-∞, ∞)

  (f ⋅ g)(x) = (x / (x + 1)) ⋅ (x - 1)

  Domain: (-∞, -1) ∪ (-1, ∞) ∩ (-∞, ∞) = (-∞, -1) ∪ (-1, ∞)

3. Division: (f / g)(x) = f(x) / g(x)

The domain of (f / g)(x) is the intersection of the domains of f(x) and g(x), excluding any values of x for which g(x) = 0.

Example:

• f(x) = x² Domain: (-∞, ∞)

• g(x) = x - 3 Domain: (-∞, 3) ∪ (3, ∞)

  (f / g)(x) = x² / (x - 3)

  Domain: (-∞, ∞) ∩ (All real numbers except 3) = (-∞, 3) ∪ (3, ∞)

4. Composition of Functions: (f ∘ g)(x) = f(g(x))

The domain of (f ∘ g)(x) consists of all x in the domain of g(x) such that g(x) is in the domain of f(x).

1. Find the domain of g(x).
2. Find the range of g(x).
3. Find the domain of f(x).
4. Ensure that the range of g(x) is a subset of the domain of f(x). If not, restrict the domain of g(x) accordingly.

Example:

• f(x) = √x Domain: [0, ∞)

• g(x) = x + 1 Domain: (-∞, ∞)

  (f ∘ g)(x) = f(g(x)) = √(x + 1)

  For √(x + 1) to be defined, x + 1 ≥ 0, so x ≥ -1.

  Domain: [-1, ∞)

Piecewise Functions

Piecewise functions are defined by different rules for different intervals of their domain.

Example:

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

To find the domain of a piecewise function, combine the intervals for which each piece is defined.

In this example, the function is defined for x < 0 and x ≥ 0, which covers all real numbers.

Domain: (-∞, 0) ∪ [0, ∞) = (-∞, ∞)

Conclusion

Determining functions and their domains is a fundamental skill in mathematics. By understanding the types of functions, their rules, and the restrictions on their input values, you can accurately define and work with them. Remember to consider division by zero, even-indexed radicals, and logarithms of non-positive numbers when finding the domain. Practice with various examples, and you'll become proficient in identifying functions and determining their domains. Through this understanding, you'll build a strong foundation for more advanced mathematical concepts.