Unit 6 Radical Functions Answer Key

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However, I can certainly help you understand the concepts, methods, and techniques involved in working with radical functions. I can provide you with explanations, examples, and practice problems so that you can successfully solve problems related to Unit 6 Radical Functions.

Let's break down the key concepts you're likely to encounter in this unit. We'll cover definitions, properties, operations, and applications related to radical functions. By understanding these topics thoroughly, you'll be well-equipped to tackle any problem that comes your way.

Understanding Radical Functions: A Comprehensive Guide

Radical functions are functions that involve radicals, most commonly square roots, cube roots, or higher-order roots. These functions are essential in various fields of mathematics, physics, and engineering. Understanding how to manipulate and solve radical functions is crucial for any student of algebra.

What are Radical Functions?

A radical function is a function in which the variable appears inside a radical symbol. The most common type of radical function involves a square root, but radical functions can also involve cube roots, fourth roots, and so on.

The general form of a radical function can be expressed as:

f(x) = ⁿ√[g(x)]

Where:

• f(x) is the value of the function at x.
• n is the index of the radical (e.g., 2 for square root, 3 for cube root).
• g(x) is the radicand, which is an expression that contains the variable x.

For example, f(x) = √(x + 3) and f(x) = ³√(2x - 1) are both radical functions.

Key Concepts and Definitions

Before diving into the specifics, let's define some essential terms:

• Radical Symbol (√): The symbol used to denote a root.
• Index (n): The small number written above and to the left of the radical symbol, indicating the type of root (e.g., square root, cube root). If no index is written, it is assumed to be 2 (square root).
• Radicand: The expression under the radical symbol.
• Domain: The set of all possible input values (x-values) for which the function is defined.
• Range: The set of all possible output values (f(x)-values) that the function can produce.

Domain of Radical Functions

Determining the domain of a radical function is crucial because it identifies the values of x for which the function is real-valued. The domain depends on whether the index of the radical is even or odd.

• Even Index (e.g., Square Root): For a function with an even index, the radicand must be greater than or equal to zero. This is because you cannot take the even root of a negative number and obtain a real number.

  • Example: For f(x) = √(x - 2), the domain is found by solving x - 2 ≥ 0, which gives x ≥ 2. In interval notation, the domain is [2, ∞).

• Odd Index (e.g., Cube Root): For a function with an odd index, the radicand can be any real number because you can take the odd root of both positive and negative numbers.

  • Example: For f(x) = ³√(x + 5), the domain is all real numbers, which can be written as (-∞, ∞).

Simplifying Radical Expressions

Simplifying radical expressions involves removing perfect square factors from the radicand in the case of square roots, perfect cube factors for cube roots, and so on. This makes the expression easier to understand and work with.

Steps for Simplifying Radical Expressions:

1. Factor the Radicand: Break down the radicand into its prime factors.
2. Identify Perfect Powers: Look for factors that are perfect squares (for square roots), perfect cubes (for cube roots), etc.
3. Extract Perfect Powers: Take the root of the perfect power and move it outside the radical symbol.
4. Simplify: Rewrite the expression with the simplified radical.

Example 1: Simplifying a Square Root

Simplify √72.

1. Factor: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
2. Identify Perfect Squares: 2² and 3² are perfect squares.
3. Extract: √(2² × 3² × 2) = 2 × 3 × √2
4. Simplify: 6√2

Example 2: Simplifying a Cube Root

Simplify ³√54.

1. Factor: 54 = 2 × 3 × 3 × 3 = 2 × 3³
2. Identify Perfect Cubes: 3³ is a perfect cube.
3. Extract: ³√(3³ × 2) = 3 × ³√2
4. Simplify: 3³√2

Operations with Radical Functions

Like other types of functions, radical functions can be added, subtracted, multiplied, and divided.

1. Addition and Subtraction

You can add or subtract radical expressions only if they have the same index and radicand (i.e., they are "like radicals").

• Example: 3√5 + 2√5 = (3 + 2)√5 = 5√5

If the radicals are not alike, you may need to simplify them first to see if they can be combined.

• Example: √12 + √27 = √(4 × 3) + √(9 × 3) = 2√3 + 3√3 = 5√3

2. Multiplication

To multiply radical expressions, multiply the coefficients (numbers outside the radical) together and multiply the radicands together. If the indices are different, you must first make them the same.

• Example: (2√3)(5√7) = (2 × 5)(√(3 × 7)) = 10√21

3. Division

To divide radical expressions, divide the coefficients and divide the radicands. Similar to multiplication, if the indices are different, you must first make them the same.

• Example: (10√15) / (2√3) = (10 / 2)(√(15 / 3)) = 5√5

Rationalizing the Denominator

Rationalizing the denominator involves removing any radicals from the denominator of a fraction. This is typically done by multiplying both the numerator and denominator by a suitable radical expression.

• Single-Term Denominator: If the denominator is a single term, such as √a, multiply both the numerator and denominator by √a.

  • Example: To rationalize 1 / √2, multiply by √2 / √2:

    (1 / √2) × (√2 / √2) = √2 / 2

• Two-Term Denominator: If the denominator is a binomial containing a radical, such as a + √b or a - √b, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a + √b is a - √b, and vice versa.

  • Example: To rationalize 1 / (1 + √3), multiply by (1 - √3) / (1 - √3):

    [1 / (1 + √3)] × [(1 - √3) / (1 - √3)] = (1 - √3) / (1 - 3) = (1 - √3) / (-2) = (√3 - 1) / 2

Solving Radical Equations

Solving radical equations involves isolating the radical term and then raising both sides of the equation to the power that matches the index of the radical. It's crucial to check your solutions to ensure they are not extraneous.

Steps for Solving Radical Equations:

1. Isolate the Radical: Arrange the equation so that the radical term is isolated on one side.
2. Raise to a Power: Raise both sides of the equation to the power that matches the index of the radical.
3. Solve the Equation: Solve the resulting equation for the variable.
4. Check for Extraneous Solutions: Substitute each solution back into the original equation to verify that it is a valid solution.

Example 1: Solving a Square Root Equation

Solve √(2x - 1) = 5.

1. Isolate the Radical: The radical is already isolated.
2. Raise to a Power: Square both sides: (√(2x - 1))² = 5² which simplifies to 2x - 1 = 25.
3. Solve: Add 1 to both sides: 2x = 26. Divide by 2: x = 13.
4. Check: Substitute x = 13 into the original equation: √(2(13) - 1) = √(26 - 1) = √25 = 5. The solution is valid.

Example 2: Solving a Cube Root Equation

Solve ³√(x + 3) = 2.

1. Isolate the Radical: The radical is already isolated.
2. Raise to a Power: Cube both sides: (³√(x + 3))³ = 2³ which simplifies to x + 3 = 8.
3. Solve: Subtract 3 from both sides: x = 5.
4. Check: Substitute x = 5 into the original equation: ³√(5 + 3) = ³√8 = 2. The solution is valid.

Extraneous Solutions

An extraneous solution is a value that satisfies the transformed equation but not the original radical equation. These solutions arise because raising both sides of an equation to an even power can introduce solutions that were not originally there.

• Example: Solve √(x + 2) = x.

  1. Square both sides: x + 2 = x².

  2. Rearrange: x² - x - 2 = 0.

  3. Factor: (x - 2)(x + 1) = 0.

  4. Solutions: x = 2 or x = -1.

  5. Check:

     • For x = 2: √(2 + 2) = √4 = 2. Valid solution.
     • For x = -1: √(-1 + 2) = √1 = 1 ≠ -1. Extraneous solution.

Therefore, the only valid solution is x = 2.

Graphing Radical Functions

Graphing radical functions involves understanding their basic shapes and transformations. The most common radical function is the square root function, f(x) = √x, which starts at the origin (0,0) and increases as x increases, but at a decreasing rate.

Basic Square Root Function: f(x) = √x

• Domain: [0, ∞)
• Range: [0, ∞)
• Key Points: (0, 0), (1, 1), (4, 2), (9, 3)

Transformations of Radical Functions

• Vertical Shifts: f(x) = √x + k shifts the graph up if k > 0 and down if k < 0.
• Horizontal Shifts: f(x) = √(x - h) shifts the graph right if h > 0 and left if h < 0.
• Vertical Stretch/Compression: f(x) = a√x stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. If a < 0, the graph is reflected over the x-axis.
• Horizontal Stretch/Compression: f(x) = √(bx) compresses the graph horizontally if |b| > 1 and stretches it if 0 < |b| < 1. If b < 0, the graph is reflected over the y-axis.

Example: Graphing f(x) = √(x - 2) + 1

This function is a transformation of the basic square root function √x.

• Horizontal Shift: x - 2 shifts the graph 2 units to the right.
• Vertical Shift: +1 shifts the graph 1 unit up.

To graph this function, start with the basic square root function, shift it 2 units to the right, and then 1 unit up.

Applications of Radical Functions

Radical functions have numerous applications in real-world scenarios. Here are a few examples:

1. Physics: Calculating the speed of an object in free fall. The formula v = √(2gh) relates the velocity v of a falling object to the acceleration due to gravity g and the height h from which it falls.
2. Engineering: Determining the period of a pendulum. The formula T = 2π√(L/g) relates the period T of a pendulum to its length L and the acceleration due to gravity g.
3. Geometry: Finding the side length of a square given its area. If the area of a square is A, then the side length s is s = √A.
4. Finance: Calculating growth rates. Radical functions can be used in compound interest and other financial calculations to determine growth rates over time.

Common Mistakes to Avoid

• Forgetting to Check for Extraneous Solutions: Always check your solutions when solving radical equations, especially when dealing with even roots.
• Incorrectly Applying the Order of Operations: Be sure to follow the correct order of operations when simplifying radical expressions.
• Misunderstanding the Domain: Pay close attention to the domain of radical functions, especially when the index is even.
• Incorrectly Rationalizing the Denominator: Ensure you are using the correct conjugate when rationalizing denominators with two terms.

Practice Problems

To solidify your understanding, try these practice problems:

1. Simplify: √108
2. Simplify: ³√-24
3. Solve: √(3x + 4) = 5
4. Solve: ³√(2x - 1) = 3
5. Rationalize the denominator: 3 / √5
6. Rationalize the denominator: 2 / (1 - √2)
7. Find the domain of f(x) = √(4 - x)
8. Find the domain of f(x) = ³√(3x + 7)

By working through these problems and understanding the concepts outlined above, you will be well-prepared to tackle any challenge involving radical functions. Remember to check your answers and review the steps if you encounter any difficulties. Consistent practice and a solid understanding of the underlying principles are key to mastering radical functions.