Questions: Find the domain and intercepts. f(x) = sqrt(x+81) 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 [-81, ∞). (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 = -81. (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 = . (Simplify your answer. Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is no y-intercept.

Find the domain and intercepts. f(x) = sqrt(x+81) 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 [-81, ∞). (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 = -81. (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 = . (Simplify your answer. Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is no y-intercept.
Transcript text: Find the domain and intercepts. \[ f(x)=\sqrt{x+81} \] 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 $[-81, \infty)$. (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) $x=-81$. (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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Solution

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Solution Steps

To find the domain of the function \( f(x) = \sqrt{x+81} \), we need to determine the values of \( x \) for which the expression under the square root is non-negative. For the x-intercept, we set \( f(x) = 0 \) and solve for \( x \). For the y-intercept, we evaluate \( f(x) \) at \( x = 0 \).

Step 1: Find the Domain

To find the domain of the function \( f(x) = \sqrt{x + 81} \), we require the expression under the square root to be non-negative: \[ x + 81 \geq 0 \] Solving this inequality gives: \[ x \geq -81 \] Thus, the domain in interval notation is: \[ [-81, \infty) \]

Step 2: Find the x-intercept

To find the x-intercept, we set \( f(x) = 0 \): \[ \sqrt{x + 81} = 0 \] Squaring both sides results in: \[ x + 81 = 0 \implies x = -81 \] Therefore, the x-intercept is: \[ (-81, 0) \]

Step 3: Find the y-intercept

To find the y-intercept, we evaluate \( f(x) \) at \( x = 0 \): \[ f(0) = \sqrt{0 + 81} = \sqrt{81} = 9 \] Thus, the y-intercept is: \[ (0, 9) \]

Final Answer

The answers are:

  • The domain is \(\boxed{[-81, \infty)}\).
  • The x-intercept is \(\boxed{-81}\).
  • The y-intercept is \(\boxed{9}\).
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