Sraph the function. Give the domain and range.
(x)

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Nov 05,2024

Question

Sraph the function. Give the domain and range. (x) $$ =\log _{1 / 2}(\mathrm{x}-2) $$

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = \log_{1/2}(x - 2) $$, we need to determine its domain and range.

### Domain
The logarithmic function $$ \log_b(a) $$ is defined only for $$ a > 0 $$. Therefore, we need to find when the argument of the logarithm, $$ x - 2 $$, is greater than zero:

$$
x - 2 > 0
$$

Solving this inequality:

$$
x > 2
$$

Thus, the domain of the function is:

$$
\text{Domain: } (2, \infty)
$$

### Range
The function $$ f(x) = \log_{1/2}(x - 2) $$ is a logarithm with a base of $$ \frac{1}{2} $$, which is less than 1. This means the function is decreasing.

As $$ x $$ approaches $$ 2 $$ from the right (i.e., $$ x \to 2^+ $$), $$ x - 2 $$ approaches $$ 0 $$, and thus $$ f(x) $$ approaches $$ -\infty $$:

$$
\lim_{x \to 2^+} f(x) = \log_{1/2}(0) = -\infty
$$

As $$ x $$ approaches $$ \infty $$, $$ x - 2 $$ also approaches $$ \infty $$, and thus $$ f(x) $$ approaches $$ \infty $$:

$$
\lim_{x \to \infty} f(x) = \log_{1/2}(\infty) = \infty
$$

Since the function is decreasing and covers all values from $$ -\infty $$ to $$ \infty $$, the range of the function is:

$$
\text{Range: } (-\infty, \infty)
$$

### Summary
- **Domain:** $$ (2, \infty) $$
- **Range:** $$ (-\infty, \infty) $$
- Domain: $$ (2, \infty) $$ 	- Range: $$ (-\infty, \infty) $$

Sraph the function. Give the domain and range.
(x)

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