Write the domain in interval notation.
(a)
(b)
©

Ask by Montgomery Pierce. in the United States

Jan 23,2025

Question

Write the domain in interval notation. (a) $$ a(x)=\sqrt{x+16} $$ (b) $$ b(x)=\sqrt{16-x} $$ (c) $$ c(x)=\frac{1}{\sqrt{x+16}} $$

UpStudy AI · Accepted Answer

Sure, let's determine the domain for each function by analyzing the expressions inside the square roots and ensuring that they are defined (i.e., the expressions under the square roots are non-negative and denominators are not zero where applicable).

### (a) $$ a(x) = \sqrt{x + 16} $$

**Condition:**  
The expression inside the square root must be non-negative:
$$ x + 16 \geq 0 $$

**Solution:**  
$$ x \geq -16 $$

**Domain:**  
$$ [-16, \infty) $$

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### (b) $$ b(x) = \sqrt{16 - x} $$

**Condition:**  
The expression inside the square root must be non-negative:
$$ 16 - x \geq 0 $$

**Solution:**  
$$ x \leq 16 $$

**Domain:**  
$$ (-\infty, 16] $$

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### (c) $$ c(x) = \frac{1}{\sqrt{x + 16}} $$

**Conditions:**  
1. The expression inside the square root must be positive (since it's in the denominator and cannot be zero):
$$ x + 16 > 0 $$
2. The denominator cannot be zero.

**Solution:**  
$$ x > -16 $$

**Domain:**  
$$ (-16, \infty) $$

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**Summary of Domains:**

- **(a)** $$[ -16,\ \infty )$$
- **(b)** $$( -\infty,\ 16 ]$$
- **(c)** $$( -16,\ \infty )$$
- (a) $$[-16, \infty)$$ - (b) $$(-\infty, 16]$$ - (c) $$(-16, \infty)$$

Write the domain in interval notation.
(a)
(b)
©

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Write the domain in interval notation. (a) (b) ©