Find the domain and intercepts \[ f(x)=\sqrt{x+36} \] Find the domain. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of the function \( f(x) \) is \( [-36, \infty) \). (Type your answer in interval notation.) B. The domain is all real numbers. Find the x-intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) of the graph is (are) \( x=-36 \). (Simplify your answer. Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is no \( x \)-intercept. Find the \( y \)-intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercept(s) of the graph is (are) \( y= \) \( \square \) (Simplify your answer. Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is no \( y \)-intercept
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The function \( f(x) = \sqrt{x + 36} \) has a specific domain where the expression under the square root must be non-negative. This means that \( x + 36 \geq 0 \) or \( x \geq -36 \). Thus, the domain of the function \( f(x) \) is \( [-36, \infty) \), which means the graph of the function starts at \( -36 \) on the x-axis and goes infinitely to the right. For the intercepts, the x-intercept occurs when \( f(x) = 0 \). Setting \( \sqrt{x + 36} = 0 \) leads to \( x + 36 = 0 \), giving us \( x = -36 \) as the only x-intercept. Since this is the point where the graph touches the x-axis, it's a key feature of the function. For the y-intercept, we evaluate \( f(0) \), leading to \( f(0) = \sqrt{36} = 6 \), so the graph crosses the y-axis at \( y = 6 \).
