$$ \sqrt{49-x^2} $$

Find the domain and range of the function. (Enter your answers using interval notation.)


Given the function $ f(x) = \sqrt{49 - x^2} $, we need to determine its domain and range.

### Domain:
For the square root function to be defined, the expression inside the square root must be non-negative. Therefore, we require:

\[
49 - x^2 \geq 0
\]

This inequality can be solved as follows:

1. Rearrange the inequality:
   \[
   49 \geq x^2
   \]

2. Take the square root of both sides (considering both positive and negative roots):
   \[
   -7 \leq x \leq 7
   \]

This means $ x $ must lie within the interval $[-7, 7]$.

Thus, the domain is:
\[
\text{Domain: } [-7, 7]
\]

### Range:
The range of the function depends on the values that $ f(x) $ can take as $ x $ varies within the domain.

- The maximum value of $ f(x) = \sqrt{49 - x^2} $ occurs when $ x = 0 $ because $ 49 - 0^2 = 49 $, and $ \sqrt{49} = 7 $.
- The minimum value of $ f(x) $ occurs at the endpoints of the domain ($ x = -7 $ or $ x = 7 $), where $ 49 - 49 = 0 $, and $ \sqrt{0} = 0 $.

Therefore, $ f(x) $ can take any value between 0 and 7.

Thus, the range is:
\[
\text{Range: } [0, 7]
\]

### Final Answer:
- **Domain:** $[-7, 7]$
- **Range:** $[0, 7]$

Would you like more details on solving inequalities involving square roots?

---

**Related Questions:**

1. How do you find the domain and range of $ f(x) = \sqrt{x^2 - 49} $?
2. What is the domain and range of $ f(x) = \sqrt{25 - x^2} $?
3. How can you solve $ \sqrt{x^2 - 4} \geq 0 $ to find the domain?
4. What happens to the domain and range if the function is $ f(x) = 7 - \sqrt{49 - x^2} $?
5. How do transformations (shifts, stretches) affect the domain and range of square root functions?

**Tip:** For square root functions, always check that the expression under the square root is non-negative to determine the domain.

Explore the domain and range of the function sqrt(49 - x^2) with detailed step-by-step solutions. Understand how to determine where the function is defined and the range of possible output values. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation