Glossary

8.7 Use Radicals in Functions

3 min read•june 25, 2024

Radical functions introduce a new level of complexity to algebraic equations. They involve roots and can have restricted domains, making them both challenging and intriguing. Understanding how to solve, , and analyze these functions is crucial for mastering more advanced mathematical concepts.

Radicals in functions require careful consideration of restrictions and simplification techniques. By learning to solve, graph, and determine domains of radical functions, you'll gain valuable skills for tackling more complex mathematical problems in future studies and real-world applications.

Radicals in Functions

Solving radical functions

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  • Substitute the given input value for the variable in the
    • Plug in the input value wherever the variable appears in the
    • f(x)=x+1f(x) = \sqrt{x + 1}, find f(3)f(3)
      • Replace x with 3: f(3)=3+1=4=2f(3) = \sqrt{3 + 1} = \sqrt{4} = 2
  • Simplify the resulting expression by evaluating the radical
    • For square roots, find the (non-negative) of the (expression under the radical symbol)
    • For cube roots, determine the real number that, when cubed, equals the radicand
    • For higher-order roots (4th, 5th, etc.), use a calculator or approximate the value

Domain of radical functions

  • Consists of all input values for which the radical function is defined
  • For (square roots, 4th roots), the radicand must be non-negative
    • f(x)=x−2f(x) = \sqrt{x - 2}, the domain is x≥2x \geq 2
      • Radicand x−2x - 2 must be non-negative: x−2≥0x - 2 \geq 0
      • Solve : x≥2x \geq 2
  • For (cube roots, 5th roots), the domain is all real numbers
    • g(x)=x+13g(x) = \sqrt[3]{x + 1}, the domain is all real numbers
  • Exclude values that make the radicand undefined (division by zero, negative even- radicands)

Graphs of radical functions

  • Determine the domain of the radical function
  • Calculate points by choosing x-values from the domain and finding corresponding y-values
    • f(x)=xf(x) = \sqrt{x}, domain: x≥0x \geq 0
      1. Select x-values: 0, 1, 4, 9
      2. Calculate y-values: f(0)=0f(0) = 0, f(1)=1f(1) = 1, f(4)=2f(4) = 2, f(9)=3f(9) = 3
      3. Plot points: (0, 0), (1, 1), (4, 2), (9, 3)
  • Connect the plotted points with a smooth curve
  • Identify key features of the graph
    • : point where graph crosses y-axis
    • (s): point(s) where graph crosses x-axis
    • Increasing/decreasing behavior: whether function rises or falls as x increases
    • Domain: set of all possible x-values
    • : set of all possible y-values (output values of the function)
  • Interpret the graph's meaning in the context of the problem, if applicable

Functions, Graphs, and Inequalities

  • A function is a rule that assigns each input value to exactly one output value
  • The graph of a function is a visual representation of all its input-output pairs
  • An inequality in a radical function represents a relationship between expressions involving radicals
    • Example: x+2>3\sqrt{x + 2} > 3 (solve by squaring both sides and considering the domain)
  • The domain and range of a radical function can often be determined by analyzing its graph

Key Terms to Review (26)

Cube Root: The cube root is a mathematical operation that finds the value that, when multiplied by itself three times, results in the original number. It is the inverse operation of raising a number to the power of three.
Cube Root (∛): The cube root, denoted by the symbol ∛, is a mathematical operation that finds the value that, when multiplied by itself three times, equals the original number. It is one of the fundamental operations in algebra and is closely related to the concepts of exponents and radicals.
Domain: The domain of a function refers to the set of all possible input values for that function. It represents the range of values that the independent variable can take on, and it determines the set of values for which the function is defined.
Even-Index Radicals: Even-index radicals refer to square roots and higher even-powered roots, such as fourth roots or sixth roots. These radicals involve raising a number or expression to an even-numbered power and then taking the principal root of that result.
Exponent-Radical Duality: Exponent-radical duality refers to the mathematical relationship between exponents and radicals, where they can be used interchangeably to represent the same underlying concept. This duality allows for flexible and efficient manipulation of expressions involving both exponents and radicals in the context of functions.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions describe how changes in one quantity affect changes in another quantity.
Graph: A graph is a visual representation of the relationship between variables, often used to depict the behavior of a function. Graphs can take various forms, such as line graphs, scatter plots, or bar charts, and are essential tools for analyzing and understanding mathematical and scientific data.
Index: The index of a radical expression is the number that indicates the root being taken. It specifies the root, such as a square root, cube root, or fourth root, and is used to simplify and manipulate radical expressions.
Inequality: Inequality is a mathematical relationship between two quantities where one is greater or less than the other. It is a fundamental concept in mathematics that extends beyond the simple equality of values and allows for the comparison and analysis of different magnitudes.
Non-Negative Radicand: A non-negative radicand is a value inside a radical expression that is greater than or equal to zero. This is an important concept in the context of using radicals in functions, as the radicand must be non-negative for the radical to be defined and the function to be valid.
Nth Root: The nth root of a number is the value that, when raised to the power of n, equals the original number. It represents the inverse operation of exponentiation, where the nth root extracts the value that was raised to the power of n.
Odd-Index Radicals: Odd-index radicals are square roots, cube roots, and other roots with odd indices, such as the fifth root or seventh root. These types of radicals are commonly used in functions and equations to represent non-perfect squares, cubes, or higher-order powers.
Principal Square Root: The principal square root of a non-negative real number is the non-negative real number that, when multiplied by itself, gives the original number. It is a fundamental concept in mathematics, particularly in the context of using radicals in functions.
Product Property of Radicals: The product property of radicals states that the square root of the product of two numbers is equal to the product of their square roots. This property is fundamental in simplifying and manipulating radical expressions involving multiplication.
Quotient Property of Radicals: The quotient property of radicals is a fundamental rule that allows for the simplification of radical expressions involving division. It states that the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.
Radical Equations: Radical equations are mathematical equations that contain one or more square roots or other root functions. These equations must be solved using specialized techniques to isolate and simplify the radical terms in order to find the solution(s).
Radical Expression: A radical expression is a mathematical expression that contains one or more square roots or higher-order roots. These expressions represent the inverse operation of raising a number to a power, and they are used to represent values that cannot be expressed as a simple integer or fraction.
Radical Function: A radical function is a type of function that contains a square root or other root as a part of its expression. These functions are characterized by their non-linear and often non-continuous nature, with distinct behavior around the root values.
Radical Inequalities: Radical inequalities are mathematical expressions that involve inequalities with variables under radical signs. These types of inequalities require specialized techniques to solve and analyze, as the presence of radicals introduces additional complexity compared to standard linear or polynomial inequalities.
Radicand: The radicand is the quantity or expression under the radical sign in a radical expression. It represents the value or number that is to be operated on by the radical symbol, such as the square root or cube root.
Range: The range of a set of data or a function is the difference between the largest and smallest values in the set. It represents the spread or variation within the data and is a measure of the dispersion or variability of the values.
Rationalizing Denominators: Rationalizing denominators is the process of eliminating radical expressions from the denominator of a fraction, making the denominator a rational number. This is an important technique in simplifying radical expressions and functions involving radicals.
Simplifying Radicals: Simplifying radicals is the process of reducing a radical expression to its simplest form by removing any perfect squares from the radicand and rationalizing the denominator if necessary. This is an essential skill in algebra that allows for more efficient manipulation and calculation of radical expressions.
Square Root: The square root, denoted by the symbol √, is a mathematical operation that represents the inverse of squaring. It is used to find the value that, when multiplied by itself, results in the original number. The square root of a number is the value that, when raised to the power of 2, equals the original number.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation crosses the x-axis. It represents the value of x when the function's y-value is zero, indicating the horizontal location where the graph intersects the horizontal axis.
Y-intercept: The y-intercept is the point where a line or graph intersects the y-axis, representing the value of the function when the independent variable (x) is equal to zero. It is a crucial concept in understanding the behavior and characteristics of various types of functions and their graphical representations.
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