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Chapter 10: Problem 94

Determine the domain of each function described. $$ g(x)=\sqrt{x+8} $$

Short Answer

Expert verified
The domain of the function is \( [ -8, \infty ) \).

Step by step solution

01

Identify the function type

The function given is in the form of a square root, which is written as \( g(x)=\sqrt{x+8} \). The domain of a square root function is determined by the condition that the expression under the square root must be greater than or equal to zero.
02

Set up the inequality

To find the domain, set up the inequality from the expression under the square root. Here, the expression is \( x + 8 \). Thus, we need to solve the inequality \( x + 8 \geq 0 \).
03

Solve the inequality

Solve the inequality \( x + 8 \geq 0 \). Subtract 8 from both sides of the inequality to get \( x \geq -8 \).
04

Write the domain

The domain of the function \( g(x) = \sqrt{x+8} \) is all real numbers \( x \) such that \( x \geq -8 \). In interval notation, this is written as \( [ -8, \infty ) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The function in the exercise is a square root function, which is commonly written in the form of \( g(x) = \sqrt{expression} \). One of the key properties of the square root function is that the expression under the square root must be non-negative, because the square root of a negative number is not real in the context of real numbers.

A square root function graphically represents half of a parabola that opens upwards and to the right.

For example, if we have \( g(x) = \sqrt{x + 8} \), the expression under the square root is \( x + 8 \). To determine the domain, you need to make sure \( x + 8 \geq 0 \). This ensures the value inside the square root is zero or positive.
Inequality Solving
To find the domain of the square root function, we must solve the inequality that arises from the condition that the expression inside the square root be non-negative.

Start with the inequality \( x + 8 \geq 0 \). Solving this involves basic algebraic steps. First, isolate the variable \( x \). Subtract \( 8 \) from both sides, leading to \( x \geq -8 \).

This means any value of \( x \) that is greater than or equal to \( -8 \) makes the expression under the square root non-negative, and hence fits the function's domain.
Interval Notation
Once the inequality is solved, the solution can be expressed using interval notation, which is a concise way to describe the set of all values that satisfy the inequality.

For our inequality \( x \geq -8 \), interval notation expresses this as \( [ -8, \infty ) \). The square bracket \( [ \) indicates that \( -8 \) is included in the domain (closed interval), and the parenthesis \( ) \) indicates that the values go infinitely large and \( \infty \) is not included in the domain (open interval).

Understanding interval notation is essential for properly communicating the domain of functions, especially when it involves ranges and boundless values.

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