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!