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To find the domain and range of the function \( f(x) = \sqrt{4 - 2x} + 5 \), we need to consider the conditions under which the expression inside the square root is non-negative (since the square root function is only defined for non-negative values) and the values that the function can attain.

### 1. Domain of \( f(x) \)

The expression inside the square root, \( 4 - 2x \), must be non-negative:

\[
4 - 2x \geq 0
\]

Let's solve for \( x \):

\[
4 \geq 2x
\]

\[
2 \geq x
\]

Thus, the domain of \( f(x) \) is:

\[
x \in (-\infty, 2]
\]

### 2. Range of \( f(x) \)

To find the range, we need to analyze the possible values of \( f(x) \). The square root function \( \sqrt{4 - 2x} \) takes values from 0 to \( \sqrt{4} = 2 \) as \( x \) ranges from 2 to negative infinity. Therefore, we have:

\[
\sqrt{4 - 2x} \in [0, 2]
\]

Adding 5 to the square root, we have:

\[
f(x) = \sqrt{4 - 2x} + 5
\]

So, the range of \( f(x) \) is:

\[
f(x) \in [5, 7]
\]

### Conclusion

- **Domain**: \( (-\infty, 2] \)
- **Range**: \( [5, 7] \)

Would you like more details on any part of this process or have any specific questions?

### Related Questions:

1. What happens to the domain if the square root function is changed to a cube root?
2. How would the domain and range change if the function were \( f(x) = \sqrt{4 - 2x} - 5 \)?
3. How do we find the domain and range of a rational function?
4. Can the range be determined graphically, and how would the graph of this function look?
5. What if the function were \( f(x) = 4 - 2x + 5 \) without a square root? What would its domain and range be?

### Tip:
Always remember to check the domain restrictions first when dealing with square roots, logarithms, or rational functions!

Find the domain and range for f(x) = √(4 - 2x) + 5.

Learn how to find the domain and range of the function f(x) = √(4 - 2x) + 5. This problem covers key concepts in algebra, including determining the domain of a square root function and finding the range of the function. Suitable for students in Grades 9-12.

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