Restricted Domain Math Example 3

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Example 3

medium

Find the domain of

g(x) = \frac{\sqrt{x+4}}{x^2 - 9}

Solution

1
Requirement 1: $x + 4 \ge 0 \Rightarrow x \ge -4$ . Requirement 2: $x^2 - 9 \neq 0 \Rightarrow x \neq \pm 3$ .
2
Combine: $x \ge -4$ and $x \neq 3$ and $x \neq -3$ . Domain: $[-4, -3) \cup (-3, 3) \cup (3, \infty)$ .

Answer

[-4, -3) \cup (-3, 3) \cup (3, \infty)

When a function combines multiple operations, each imposes its own restriction. The overall domain is the intersection of all individual restrictions. Here, the square root requires non-negative input and the fraction requires a nonzero denominator.

About Restricted Domain

Restricting a domain limits allowable inputs so a function has desired properties, often invertibility.

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More Restricted Domain Examples

Example 1 easy

Find the natural (implied) domain of [formula].

Example 2 medium

Restrict the domain of [formula] so that the function has an inverse. Find the inverse on this restr

Example 4 hard

Restrict the domain of [formula] to the largest interval containing [formula] on which [formula] is

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Related Concepts

Domain Function Inverse Function