Inverse Trig Functions, finding Domain and Range

Question

I understand the restricted domains of inverse trig functions, but what about:

http://www4c.wolframalpha.com/Calculate/MSP/MSP15401hd5hihhd9h8aaaa00004fd19hd4093dd23e?MSPStoreType=image/gif&s=61&w=124.&h=18.

I don't quite understand how to find the domain and range of this function.

Answer 1

Domain = $\{x \colon -1 \leq 3x - 4 \leq 1 \}$

Range = $\{3y+2 \colon 0\leq y\leq \pi\}$

Answer 2

For the domain, we know that in $\cos^{-1}(t)$, we must have $-1 \leq t \leq 1$. So in our given expression we need $-1 \leq 3x-4 \leq 1$. We can solve this for $x$: $$ -1 \leq 3x-4 \leq 1 \\ 3 \leq 3x \leq 5 \\ 1 \leq x \leq \frac{5}{3} \\ $$

Similarly for the range, $\cos^{-1}(t)$ returns a number between $0$ and $\pi$ for whatever $t$ is. We can use this to find the range of the above expression (just put $3x-4$ in place of $t$): $$ 0 \leq \cos^{-1}(3x-4) \leq \pi \\ 0 \leq 3\cos^{-1}(3x-4) \leq 3\pi \\ 2 \leq 3\cos^{-1}(3x-4)+2 \leq 3\pi+2 $$

Answer 3

To find the domain of the function, you should then be able to solve $3x-4=1 $ and $3x-4 =1$. Therefore for the minimum x value, $x=1$ and the maximum value is equal to $x=5/3$. The range is the maximum y value given the domain in this case.