Find the domain and range of the function 2 y  f (x)  64  x (4 Marks)

The given function is:

\[
y = f(x) = \sqrt{64 - x}
\]

### Domain:

The domain of the function consists of all the values of \(x\) for which the function is defined.

- The square root function \(\sqrt{64 - x}\) is only defined when the expression inside the square root is non-negative (i.e., \(64 - x \geq 0\)).
- Solving \(64 - x \geq 0\) for \(x\):

\[
64 \geq x \quad \text{or} \quad x \leq 64
\]

- Therefore, the domain of the function is all real numbers \(x\) such that \(x \leq 64\).

So, the domain is:

\[
\text{Domain: } (-\infty, 64]
\]

### Range:

The range of the function consists of all the possible values of \(y\) (or \(f(x)\)) that the function can take.

- Since \(f(x) = \sqrt{64 - x}\) is a square root function, \(f(x)\) will always be non-negative.
- The minimum value of \(f(x)\) occurs when \(x = 64\):

\[
f(64) = \sqrt{64 - 64} = \sqrt{0} = 0
\]

- The maximum value of \(f(x)\) occurs when \(x\) approaches \(-\infty\), but the practical maximum is at \(x = 0\):

\[
f(0) = \sqrt{64 - 0} = \sqrt{64} = 8
\]

- Since \(f(x)\) decreases as \(x\) increases from \(-\infty\) to 64, the range of the function is from 0 to 8.

So, the range is:

\[
\text{Range: } [0, 8]
\]

### Summary:

- **Domain:** \((-∞, 64]\)
- **Range:** \([0, 8]\)

Would you like more details or have any questions?

### Related Questions:
1. How do you graph square root functions?
2. What are the key characteristics of square root functions?
3. How do you determine the domain of a function involving square roots?
4. What happens to the range of the function if the square root is multiplied by a negative value?
5. How do transformations (like shifts and stretches) affect the domain and range?
6. Can a square root function have a domain that includes negative values of \(x\)?
7. How do you find the inverse of a square root function?
8. What are the real-world applications of square root functions?

### Tip:
When finding the domain of a function involving a square root, always ensure the expression under the square root is non-negative. This is a critical step in ensuring the function is properly defined.

Discover how to find the domain and range of the function y = sqrt(64 - x) with detailed explanations and examples. Understand the domain as all real numbers x such that x <= 64 and the range as values y from 0 to 8. Suitable for high school students studying square root functions.

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