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

Question

𝔂 = |x| - 2

Accepted Answer

Welcome back everyone to another video. What is the domain and range of the function G of X equals negative, the absolute value of x + 5. So let's begin with the domain of this function, and let's recall that the domain of a function essentially consists of all the X values for which our function is defined. Now let's recall that the absolute value function, the absolute value of X, is a piecewise function. It consists of two linear functions. X if x is greater than or equal to 0 and negative x if X is less than 0. So because it is a linear function, well, we can take any X we want and evaluate the function. What does that mean? Well, G of X in this context is basically a transformation of the absolute value of X. Where we are performing a vertical shift right and a reflection, so it doesn't really matter, and the domain remains the same. We can take any x value to evaluate the absolute value of X from negative infinity up to infinity. So we're going to say that the domain is X belongs to the interval from negative infinity up to infinity or simply all real numbers. Now let's consider the range of the given function. And let's understand that whenever we calculate the absolute value of X, it is going to be greater than or equal to 0 for any X because the absolute value turns a number positive if it's negative. So the minimum we can get is 0, right? Because the absolute value of 0 is 0. So it's always greater than or equal to 0. Now, what if we manipulate this expression and multiply both sides by -1? We're going to get negative. The absolute value of X is going to be less than or equal to 0. We're basically flipping sign. Because we're multiplying by a negative number, and now what if we add 5 to both sides, while negative, the absolute value of X plus 5 is going to be less than or equal to 0 + 5. And this is how we got our range. We can conclude that for our range. Why is In the interval from 5 up to infinity. In this context, actually, because it's less than or equal to 5, this is our upper bound, right? So we have to correct it. So from negative infinity up to 5. Negative infinity up to 5 inclusive right because our inequalities signs us less than or equal to. Well we have our answers. X belongs to the interval from negative infinity up to infinity. That's our domain. And for our range Y belongs to the interval from negative infinity up to 5 inclusive. Thank you for watching.

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

𝔂 = |x| - 2

Key Concepts

Domain

Range

Absolute Value Function

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