Question

Find the domain of the function. \(g(x)=\sqrt{x+1}\)

Short answer

The domain of the function is \([-1, +\infty)\).

Step-by-step solution

Identify the inequality for domain

Since square root is not defined for negative numbers, set the terms within the square root greater than or equal to 0. So, make the inequality \(x+1 \geq 0\).

Solve the inequality

Subtract 1 from both sides of the inequality to isolate \(x\). The inequality becomes \(x \geq -1\).

Write the domain

The domain of the function \(g(x)\), in interval notation, is \([-1, +\infty)\), which means \(x\) is all real numbers greater than or equal to -1.

Key concepts

Understanding Square Root Functions

When dealing with square root functions, like the function presented in the exercise, it's essential to recognize that they are defined only for values that make the expression under the square root non-negative. This is because the square root of a negative number is not a real number. Consequently, when you encounter a function such as g(x) = \(\sqrt{x+1}\), the expression inside the square root, x+1, must be greater than or equal to zero.

The square root function is a type of radical function, and in general, it can only take non-negative numbers. Visually, the graph of a square root function typically resembles half of a parabola lying on its side, opening towards the right when dealing with the principal square root. It starts at the point determined as the least value in the domain and extends infinitely to the right in the positive direction.

Solving Inequalities Guide

Approaching an inequality can be similar to solving equations, but with one key difference: the direction of the inequality may change when you multiply or divide by a negative number. However, in the given exercise, the inequality is x + 1 \(\geq\) 0, which doesn't involve this complexity. The step to isolate x is correct – simply subtracting 1 from both sides of the inequality will suffice.

Understanding inequalities is crucial when you want to find the domain of a function, which is the set of all possible input values. Inequalities define a range of values that satisfy a certain condition, and in the context of domain, they help determine what that set contains. The solution for x \(\geq\) -1 represents all the real numbers greater than or equal to -1, which forms the domain of the square root function in the exercise.

Interval Notation Explained

The use of interval notation provides a compact way to represent a range of numbers along the number line. In the solution provided, the domain is expressed in interval notation as [ -1, +\(\infty\) ). Here's what this means:

The square bracket [ indicates that -1 is included in the domain – think of it as 'locked in'. Conversely, the round parenthesis ) next to +\(\infty\) tells you that infinity is not a number that can be reached or included; hence, it's 'open'. Accordingly, the domain consists of all real numbers from -1 (including -1 itself) and extends without bound to positive infinity.

Interval notation is preferred over other forms of notation for its simplicity and clarity, especially when communicating solutions in mathematics. Recognizing how to read and write in this notation is an essential skill for interpreting and defining domains and ranges of functions.