# What is the domain and range of ln(1-x^2)?

Jun 6, 2018

Domain: $\left\{x | - 1 < x < 1\right\}$ or in interval notation $\left(- 1 , 1\right)$

Range: $\left\{y | y \le 0\right\}$ or in interval notation $\left(- \infty , 0\right]$

#### Explanation:

$\ln \left(1 - {x}^{2}\right)$

The input to the natural log function must be greater than zero:

$1 - {x}^{2} > 0$

$\left(x - 1\right) \left(x + 1\right) > 0$

$- 1 < x < 1$

Therefore Domain is:

$\left\{x | - 1 < x < 1\right\}$ or in interval notation $\left(- 1 , 1\right)$

At zero the value of this function is $\ln \left(1\right) = 0$ and as $x \to 1$ or as $x \to - 1$ the function $f \left(x\right) \to - \infty$ is the range is:

$\left\{y | y \le 0\right\}$ or in interval notation $\left(- \infty , 0\right]$

graph{ln(1-x^2) [-9.67, 10.33, -8.2, 1.8]}