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Question 13 Find the domain of \( f(x)=\sqrt{x+1} \) Use two lower case o's for infinity. "oo" is how you type in infinity. Hint: You may find it helpful to graph the function on your graphing calculator. Question Help: \( \square \) Video \( \square \) Message instructor Submit Question
Ask by Herrera Vargas. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of \( f(x) = \sqrt{x + 1} \) is all real numbers \( x \) where \( x \geq -1 \). In interval notation, this is \([-1, \text{oo})\).
Solution
To find the domain of the function \( f(x) = \sqrt{x + 1} \), we need to determine the values of \( x \) for which the expression inside the square root is non-negative (since the square root of a negative number is not a real number).
1. **Set the inside of the square root to be greater than or equal to zero:**
\[
x + 1 \geq 0
\]
2. **Solve for \( x \):**
\[
x \geq -1
\]
Therefore, the domain of \( f(x) = \sqrt{x + 1} \) includes all real numbers \( x \) such that \( x \) is greater than or equal to \(-1\).
**In interval notation, the domain is:**
\[
[-1, \text{oo})
\]
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Simplify this solution Extra Insights
To find the domain of \( f(x) = \sqrt{x+1} \), you need to ensure that the expression inside the square root is non-negative. This means \( x + 1 \geq 0 \). Solving this inequality gives \( x \geq -1 \). Therefore, the domain of \( f \) is all real numbers \( x \) such that \( x \geq -1 \), or in interval notation, \( [-1, oo) \). If you're graphing this function, you'll notice that the curve starts from the point (-1, 0) and rises to the right without bound. This visual can help reinforce the understanding that the function is defined for all \( x \) values greater than or equal to -1, emphasizing that this is where the curve begins!
