In Exercises 19–32, find the (a) domain and (b) range.
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Question

In Exercises 19–32, find the (a) domain and (b) range.

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𝔂 = -2 + √1 - x

Accepted Answer

Welcome back, everyone. What is the domain and range of the function G of X equals 5 + square root of X + 3? So let's begin with the domain of the given function. For the domain, we have to recall that the domain of a function consists of all the possible X values for which the function is defined. And because we have a radical term, we have to recall that. Whenever we take an even root of a specific function. Let's say 2 N. Representing an even root such as square root, 4th root, and so on, specifically, the term under the radical must be non-negative, right, because we cannot seek. And even root of a negative number in a real plane, essentially F of X must be greater than or equal to 0. So in this case our function is x + 3, and it must be greater than or equal to 0. It must be nonnegative. So if we solve this inequality, well, we have to subtract 3 from both sides, and this gives us X being greater than or equal to -3, meaning our domain. Consists of all the X values in the innerval from -3 inclusive up to infinity, right? This is what the inequality says. Now let's consider the range. Whenever we consider the range, well, those are basically the Y values or G values in this case. From minimum to maximum, let's recall that the square root function. Which is defined From 0 up to infinity is essentially a rising curve, right? With an increase in X, the Y value increases, so we can essentially say that this curve represents square root of X. So square root of X is defined from 0 up to infinity in terms of the y values. Now in this case we have 5 added to the radical term, so it essentially represents a vertical shift up by 5 units. So this means that the minimum value instead of being the zero for the parent function is now at 5, right? So we can also do that analytically. If we consider the radical term square root of x + 3, it is always greater than or equal to 0, right? We cannot get a negative value. So the minimum value that we can get for the radical term is 0, and then 0 + 5 gives us 0 for the lower bound of the range, we can say that our range. Is why. Belongs to the interval from. 5, that's our minimum value, right? Because we have our square root function shifted upward by 5 units and up to infinity, right? Because as we know, The square root function keeps increasing up to infinity in terms of its y value. So that's it. We have our domain and range. The domain is X being greater than or equal to -3, and our range is from 5 inclusive up to infinity. Thank you for watching.

In Exercises 19–32, find the (a) domain and (b) range.
____
𝔂 = -2 + √1 - x

Key Concepts

Domain of a Function

Range of a Function

Square Root Function

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