# How do you find the domain & range for y=7cos[3pi(x-2)]+7?

##### 1 Answer
Aug 2, 2018

Domain: $x \in \left(- \infty , \infty\right)$
Range: $\left[- 7 , 7\right]$

#### Explanation:

Any cosine value $\in \left[- 1 , 1\right]$. So,

y = 7 cos ( 3 pi ( x - 2 ) + 7 = 7 cos (3 pi x - 6 pi ) + 7

$= 7 \cos \left(3 \pi x\right) + 7 \in 7 \left[- 1 , 1\right] = \left[- 7 , 7\right]$

Amplitude of this cosine wave: 7

Period : $\frac{2 \pi}{3 \pi} = \frac{2}{3}$

Vertical shift: 7

See graph.
graph{(y-7 cos(3pix)-7)(y-7+0x)(y-14+0x)(y+0x)(x-1/3+0.0001y)(x+1/3+0.0001y)=0[-0.7 0.7 -0.6 14.1 ]}

Observe the axis shift by 7 units. The graph is not on uniform scale.