Answered

What is the domain of this radical function?

$f(x)=\sqrt{x+1}-3$


A$[3, \infty)$
B$[-3, \infty)$
C$[1, \infty)$
D$[-1, \infty)$

Verified Answer :

  • For a square root function \(\sqrt{x+1} - 3\), the radicand must satisfy the condition \(x + 1 \geq 0\).
  • Solving the inequality gives \(x \geq -1\).
  • In interval notation, this represents the set \([-1, \infty)\).
  • Therefore, the correct choice is \(\boxed{[-1, \infty)}\).

Explanation

  1. Analyzing the Function and Defining the Domain Condition

The given function is \(f(x) = \sqrt{x+1} - 3\). A square root function \(\sqrt{g(x)}\) is defined for real numbers only when the expression inside the square root, the radicand, is non-negative. This condition is necessary because the square root of a negative number is not a real number. In this problem, the radicand is \(x+1\).

  1. Solving the Inequality for the Domain

To find the domain, we must solve the inequality \(x + 1 \geq 0\). Subtracting 1 from both sides of the inequality, we get \(x \geq -1\). This means the function \(f(x)\) is defined for all values of \(x\) that are greater than or equal to -1.

  1. Determining the Correct Option

The set of all values of \(x\) satisfying \(x \geq -1\) can be written in interval notation as \([-1, \infty)\). Comparing this result to the given options:

  • \([3, \infty)\)
  • \([-3, \infty)\)
  • \([1, \infty)\)
  • \([-1, \infty)\)

The correct option is the one corresponding to \([-1, \infty)\).

Examples

The concept of a domain is crucial when working with physical measurements. For example, if you are modeling the time it takes for an object to fall a certain distance \(h\) given by the formula \(t(h) = \sqrt{\frac{2h}{g}}\) (where \(g\) is gravity), the domain of the function represents all possible heights \(h\) that make sense in this context. Since you cannot have a negative distance, the domain is \(h \geq 0\). Understanding how to calculate the domain ensures that your mathematical model accurately reflects real-world constraints.

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