Question

What is the domain and range of $\arccos \left( x-1 \right)$ ?

Answer 1

Hint: We can define the domain of a function as the complete step of possible values of the independent variable. So, we can say that the domain is the set of all possible $x$ values which will make the function ‘work’ and will give the output of $y$ as a real number. And the range of a function is the complete set of all possible resulting values of the dependent variable ($y$ usually), after we have substituted the domain.

Complete step-by-step solution:
According to our question it is asked to calculate the domain and range of $\arccos \left( x-1 \right)$ or ${{\cos }^{-1}}\left( x-1 \right)$. For determining the range and the domain we have to see the domain and the range of inverse cosine function. The range of an inverse cosine function is defined as $0\le y\le \pi $. And this is fixed for all inverse cosine functions. And the domain for an inverse cosine function must be between -1 to 1. Both the range and the domain are related to each other. The range of any function decides the values from where the function will show and the domain decides the quantity of that graph of the function.
So, if we see in the question then it is clear that it is asked to determine the domain and the range of inverse $\cos \left( x-1 \right)$. So, the range of inverse cosine functions always remains the same and that is $0\le y\le \pi $. Because these functions vary between the first and second quadrant always. So, the range is $0\le y\le \pi $ that can be written as $\left[ 0,\pi \right]$.
The domain of the inverse cosine function is -1 to 1. And this is because the values of an inverse function always varies between -1 to 1. But our function is inverse $\cos \left( x-1 \right)$. So, we have to add our domain from 1. If we increase the domain of $\arccos \left( x \right)$ by 1, then the domain is,
$-1+1\le x\le 1+1$
So, we can write it as $0\le x\le 2$.
So, the domain of $\arccos \left( x-1 \right)$ is equal to $\left[ 0,2 \right]$ and the range is $\left[ 0,\pi \right]$.

Note: We must know the trigonometry concepts to solve this question. The domain where the $x$ value ranges and the range where the $y$ value ranges. Here the function is a form of trigonometric function. So, we have to see the values in which the domain is defined that will be the value of range for this function.