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Given the equation \( y=\sqrt{x+3}+4: \) What is the domain? \( \widehat{\approx} \)

Ask by Lewis Paul. in the United States
Jan 24,2025

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Answer

The domain of the function \( y = \sqrt{x + 3} + 4 \) is all real numbers \( x \) such that \( x \geq -3 \).

Solution

To determine the **domain** of the function \( y = \sqrt{x + 3} + 4 \), we need to identify all real numbers \( x \) for which the function is **defined**. ### Step-by-Step Reasoning: 1. **Understand the Square Root Function:** - The square root function \( \sqrt{z} \) is defined **only** when the expression inside the square root (\( z \)) is **non-negative**. That is: \[ z \geq 0 \] 2. **Apply to the Given Function:** - In the function \( y = \sqrt{x + 3} + 4 \), the expression inside the square root is \( x + 3 \). Therefore, for \( y \) to be real and defined: \[ x + 3 \geq 0 \] 3. **Solve the Inequality:** \[ x + 3 \geq 0 \\ \Rightarrow x \geq -3 \] 4. **Interpretation:** - The inequality \( x \geq -3 \) tells us that \( x \) can be any real number **greater than or equal to** \(-3\). ### Conclusion: - **Domain in Inequality Form:** \[ x \geq -3 \] - **Domain in Interval Notation:** \[ [-3, \infty) \] ### Graphical Representation: On a number line, the domain extends from \(-3\) to positive infinity, including the point \(-3\): \[ \boxed{\,[-3,\ \infty)\,} \]

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To find the domain of the equation \( y=\sqrt{x+3}+4 \), we need to ensure the expression inside the square root is non-negative. This means \( x + 3 \geq 0 \), leading to \( x \geq -3 \). Therefore, the domain of the function is all real numbers \( x \) such that \( x \) is greater than or equal to \(-3\), which can be expressed in interval notation as \( [-3, \infty) \). Another fun fact about square root functions: they often have a starting point or the "corner" of their graph, which is where the expression under the square root is zero. In this case, at \( x = -3 \), the function hits its minimum point—so if you're graphing, that's where the action starts! 🌟

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