Find the domain and range of $$ y=\log _{8}(4-4 x) $$. The domain is:

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To find the domain of the function $$ y = \log_{8}(4 - 4x) $$, we need to determine the values of $$ x $$ for which the argument of the logarithm, $$ 4 - 4x $$, is positive. This is because the logarithm is only defined for positive arguments.

1. Set the argument greater than zero:
   $$
   4 - 4x > 0
   $$

2. Solve the inequality:
   $$
   4 > 4x
   $$
   $$
   1 > x
   $$
   or
   $$
   x < 1
   $$

3. Additionally, since there is no lower bound on $$ x $$ from the logarithm itself, $$ x $$ can take any value less than 1. Therefore, we also consider $$ x $$ can be any real number down to negative infinity.

Thus, the domain of the function is:
$$
(-\infty, 1)
$$

Next, we will find the range of the function $$ y = \log_{8}(4 - 4x) $$.

1. As $$ x $$ approaches $$ 1 $$ from the left (i.e., $$ x \to 1^- $$), the argument $$ 4 - 4x $$ approaches $$ 0 $$ from the positive side, and thus $$ y $$ approaches $$ -\infty $$:
   $$
   \lim_{x \to 1^-} y = \log_{8}(0^+) = -\infty
   $$

2. As $$ x $$ approaches $$ -\infty $$, the argument $$ 4 - 4x $$ becomes very large (since $$ 4x $$ becomes very negative), and thus $$ y $$ approaches $$ +\infty $$:
   $$
   \lim_{x \to -\infty} y = \log_{8}(+\infty) = +\infty
   $$

Since $$ y $$ can take any real value from $$ -\infty $$ to $$ +\infty $$, the range of the function is:
$$
(-\infty, +\infty)
$$

In summary, the domain and range of the function $$ y = \log_{8}(4 - 4x) $$ are:
- Domain: $$ (-\infty, 1) $$
- Range: $$ (-\infty, +\infty) $$
Domain: $$ (-\infty, 1) $$ Range: $$ (-\infty, +\infty) $$

Find the domain and range of \( y=\log _{8}(4-4 x) \). The domain is:

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