Question

How do you find the range of the given function with the domain D where $ g:x \to 1 - {x^2},\,D = \{ - 1,0,1\} $ ?

Answer 1

Hint: In this question, we are given a function in terms of x and we are also given a set of domains of the function and we have to find its range. For that, we must know what domain and range of function are. All the possible values that x can take, that is, the values of the x for which a function is defined is called the domain of the function. We obtain different values of the function by putting different values from the domain, thus the set of all the possible values that a function can attain is called its range. So we can easily find the range by using the above-mentioned information.

Complete step by step solution:
We are given that $ g:x \to 1 - {x^2} $ and we have to find its range when the domain is $ \{ - 1,0,1\} $ .
Let $ g:x = y $ , so we have $ y = 1 - {x^2} $
To find the range, we will put these values of x one by one in the given equation and get the corresponding value of y.
At \[x = - 1,\,y = 1 - {( - 1)^2} = 0\]
At \[x = 0,\,y = 1 - {(0)^2} = 1\]
At \[x = 1,\,\,y = 1 - {(1)^2} = 0\]
Hence the range for $ g:x \to 1 - {x^2},\,D = \{ - 1,0,1\} $ is $ R = \{ 0,1\} $ .
So, the correct answer is “ $ R = \{ 0,1\} $ ”.

Note: In the above solution, we let the function be equal to y, so x is the independent variable as it can take any value while y is the dependent variable as its value changes with the value of y. The set of values taken by x is known as the domain and the set of obtained values of y for those values of x is known as the range of the function.