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Find the domain and range of the function

$$ g(x) = \sin^{-1} (3x + 1) $$

Domain of $g(x)$ is $\left[-\frac{2}{3}, 0\right]$

Range of $g(x)$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

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Oregon State University

Harvey Mudd College

University of Nottingham

Boston College

before we find the domain and range of G of X. Let's think about the domain and range of the inverse sign function in general. So remember the inputs have to be between negative one and one, and the outputs will fall between negative pi over two and pi over two. And the graph of the inverse ein looks something like that. So that tells us that what we input into the Enver sign in this case, the three X plus one has to fall between negative one and one so we can set up in inequality. Negative one is less than or equal to three X plus. One is less than or equal to one, and we'll solve that for X. Let's subtract one from all three parts, and we get negative. Two is less than or equal to three. X is less than or equal to zero, and then we'll divide all three parts by three and we get negative. 2/3 is less than or equal to X is less than or equal to zero. So we found our domain, the interval from negative 2/3 20 Now, as long as our X values fall in that domain. We will end up with values between negative one and one to substitute into the inverse ein graph or inverse ein function. So we should still be able to get outputs between negative pi over two and pi over two. So same range is the original negative pi over to two pi over two.