# What is the range of the function ln(9-x^2)?

Nov 13, 2017

Range: $\textcolor{b l u e}{\left(- \infty , 2.197224577\right]}$ (upper value is approximate)

#### Explanation:

$\left(9 - {x}^{2}\right)$ has a maximum value of $9$
and
since $\ln \left(\ldots\right)$ is only defined for arguments $> 0$

$\textcolor{w h i t e}{\text{XXX}} \left(9 - {x}^{2}\right)$ must fall in $\left(0 , 9\right]$

${\lim}_{t \rightarrow 0} \ln \left(t\right) \rightarrow - \infty$ and (using a calculator) $\ln \left(9\right) \approx 2.197224577$

giving a range for $\ln \left(9 - {x}^{2}\right)$ of $\left(- \infty , 2.197224577\right]$