To determine the domain and range of the function $f(x) = \log_2(10 - 2x)$, let’s analyze both.

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1. Domain:

The domain is the set of all $x$-values for which the function is defined. For a logarithmic function, the argument inside the logarithm must be strictly positive.

$10 - 2x > 0$

Solve for $x$:

$-2x > -10 \quad \Rightarrow \quad x < 5$

Thus, the domain of $f(x)$ is all $x$-values less than 5.

Domain in interval notation:

$(-\infty, 5)$

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2. Range:

A logarithmic function with any positive base (like $\log_2$) has a range of all real numbers.

Thus:

Range in interval notation:

$(-\infty, \infty)$

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Final Answer:

• Domain: $(-\infty, 5)$
• Range: $(-\infty, \infty)$

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Would you like a graph or further elaboration on these points?

Here are five related questions you might find useful:

1. What happens to the function as $x$ approaches 5 from the left?
2. How does the base of the logarithm affect the shape of the function?
3. What is the behavior of $f(x)$ as $x \to -\infty$?
4. What is the value of $f(x)$ at $x = 0$?
5. How does the function transform if the base changes to $\log_{10}(10 - 2x)$?

Tip: When working with logarithmic functions, always ensure the argument is positive for the function to be defined.