Question

f(x)=x/1-x find domain and range

Answer 1

To determine the domain and range of the function \( f(x) = \frac{x}{1 - x} \), we will analyze it step by step.

### Domain:
The function \( f(x) \) is defined for all values of \( x \) except where the denominator is zero. Thus, we need to find when \( 1 - x = 0 \):
\[
1 - x = 0 \implies x = 1
\]
So, the function is undefined at \( x = 1 \). Therefore, the domain of \( f(x) \) is all real numbers except \( x = 1 \):
\[
\text{Domain: } (-\infty, 1) \cup (1, \infty)
\]

### Range:
To find the range, we can analyze the behavior of the function as \( x \) approaches the domain's endpoints and look for horizontal asymptotes.

1. **As \( x \to 1^- \) (approaching from the left)**:
\[
f(x) \to \frac{1}{1 - 1} \to -\infty
\]

2. **As \( x \to 1^+ \) (approaching from the right)**:
\[
f(x) \to \frac{1}{1 - 1} \to +\infty
\]

3. **As \( x \to -\infty \)**:
\[
f(x) \to \frac{-\infty}{1 + \infty} \to 0
\]

4. **As \( x \to +\infty \)**:
\[
f(x) \to \frac{+\infty}{1 - \infty} \to -1
\]

From this analysis, we can observe that the function does not attain the value \( -1 \) (it approaches it but never reaches it) as \( x \to \infty \). Therefore, the range of the function includes all real numbers except \( -1 \):
\[
\text{Range: } (-\infty, -1) \cup (-1, \infty)
\]

### Summary:
- **Domain**: \( (-\infty, 1) \cup (1, \infty) \)
- **Range**: \( (-\infty, -1) \cup (-1, \infty) \)