Question

Find the domain and the range for each function.$$y=\sqrt{x-1}$$

Short answer

The domain of \(y = \sqrt{x-1}\) is \([1, +\infty)\) and the range is \([0, +\infty)\).

Step-by-step solution

Identify the Domain

The domain of a function includes all the possible input values (x-values) for which the function is defined. Since the function has a square root, the expression inside the square root must be greater than or equal to zero. Thus, we need to find the values of x for which \(x - 1 \geq 0\).

Solve the Inequality

To find the domain, solve the inequality \(x - 1 \geq 0\). Adding 1 to both sides of the inequality gives us \(x \geq 1\).

Express the Domain

The domain of the function, therefore, is all real numbers greater than or equal to 1, which can be expressed in interval notation as \([1, +\infty)\).

Identify the Range

The range of a function is the set of all possible output values (y-values). Since \(y\) is the square root of a non-negative number, \(y\) will always be non-negative. Therefore, the smallest value of \(y\) is 0, when x is exactly 1.

Express the Range

Thus, the range of the function is all non-negative real numbers, which can be expressed in interval notation as \([0, +\infty)\).

Key concepts

Solving Inequalities

Understanding how to solve inequalities is crucial when finding the domain of certain types of functions. Inequalities arise when we have an expression set to be less than (<), less than or equal to (Resolving an inequality often follows the same steps as solving an equation, with some additional considerations. For instance, multiplying or dividing both sides of an inequality by a negative number flips the direction of the inequality. When you come across a problem like finding the domain of the function As observed in the exercise, to solve the inequality Knowing how to handle inequalities will not only assist in finding the domain but also play a significant role in various areas of mathematics, from algebra to calculus.

Square Root Functions

Square root functions are a type of radical function where the variable is under a square root. The standard form of a square root function is Identifying the domain of square root functions, as seen in our exercise, hinges on recognizing that the radicand (the expression under the square root) must be non-negative, because the square root of a negative number is not defined in the set of real numbers. This leads to setting up an inequality with the radicand. Square root functions have a variety of applications, especially in geometry and physics, to solve problems involving areas, distances, and other quantities that are inherently non-negative.

Interval Notation

Interval notation is a method of writing sets of numbers that describe the solution to inequalities or the domain and range of functions. It's a concise way to communicate which numbers are included in a set and whether the endpoints are included or not.The key symbols in interval notation are the parentheses, For example, the domain of our square root function is Interval notation is incredibly useful in calculus and higher-level math courses for describing intervals of continuity, differentiability, and integrability.

Function Definition

A function is a fundamental concept in mathematics where each input value is associated with exactly one output value. Mathematically, for a set X of inputs and a set Y of outputs, a function f can be defined as a relation from X to Y with the property that each element of X is related to exactly one element of Y. This relationship is often written as In the exercise provided, the square root function Properly defining functions is essential for all branches of mathematics, as it ensures clarity of thought and prevents ambiguity in problem-solving and theoretical discussions.