How do you find the domain & range for f(x)= -sin(x-π)-1 ?

Oct 26, 2015

Domain: $\xi n \mathbb{R}$
Range: $\left[0 , - 2\right] \in \mathbb{R}$

Explanation:

$\sin \left(x - \pi\right)$ has the same domain and range as $\sin \left(x\right)$; namely domain: $\mathbb{R}$; range $\left[- 1 , + 1\right]$. Subtracting $\pi$ from $x$ within the argument of $\sin$ only shifts the pattern to the left by $\pi$.

$- \sin \left(x - \pi\right)$ has the same domain and range as $\sin \left(x - \pi\right)$; the point are simply reflected in the X-axis.

$\sin \left(x - \pi\right) - 1$ has the same domain as $- \sin \left(x - \pi\right)$ but the range is reduced by $1$; that is the range becomes $\left[- 2 , 0\right]$ instead of $\left[- 1 , 1\right]$