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

Question

𝔂 = 2 + 3x² .

x² + 4

Accepted Answer

Welcome back, everyone. Determine the domain and range for the function Y equals 3 + 2 x divided by X2 + 9. Let's begin with the domain of this function. We have to recall that those are the X values for which the function is defined. And what we have to pay attention to is the second part of the function which involves the rational function, right? 2 x 2 divided by x2 + 9. For rational functions, division by 0 is undefined. So we want to ensure that x2 + 9 is not equal to 0 to make. Our function continuous right and exclude any points that make the nominator equal to zero. Now if we subtract 9 from both sides, this means that x 2d cannot be equal to -9. And no X will satisfy the equation X2 equals -9 because a square of a number is going to be greater than or equal to 0 for any X in a real plane, right? This means that we don't need to exclude any x values from the domain because our denominator is never equal to 0 and therefore we can conclude that X belongs to all real numbers. Now, what about the range? So those are the Y values. Which our function can acquire from minimum to maximum. What we're going to do is basically use the idea that X2 is greater than or equal to 0, and then what can we tell about Our fractional part of the function, right? Or basically the rational expression. If we analyze 2 X squared. Divided by X2 + 9, we can show that this fraction is always going to be greater than or equal to 0. For example, if we take X equals 0, our numerator is going to become 0, right? So this is the minimum value. And we also understand that this fraction will never be negative because our denominator is always positive, and our numerator is always non-negative, right? Because we're multiplying 2 by a non-negative number. So this is the first condition, right? We have to understand that 2 X2 divided by X2 + 9 is always greater than or equal to 0. Now, if we divide both sides of the fraction by X2, we're going to get 2 divided by. 1 + 9 divided by X squared. So from here we can also conclude that this fraction is always less than 2. 9 divided by X squared. Will not really have a minimum value as X increases infinitely. Our term approach is 0, right? So to minimize 9 divided by X2 and to maximize the fraction, we want to make sure that X approaches either positive or negative infinity. So we can say that 9. Divided by x2d approaches 0 as x approaches positive or negative infinity, right, and essentially it only approaches 0 but not equal to 0. So when we divide 2 by 1, we get 2 not inclusive. From here we can conclude that 2 x squared. Divided by X2 + 9, using the first condition and the second condition is between two values. It is greater than or equal to 0, but strictly less than 2. Why is this inequality useful? Well, essentially we can just add 3 to both sides. To get the range of the function. So we add 3. To the middle part, then we get 0 + 3, which gives us 3 on the lower end, and then 2 + 3 gives us 5. So this shows us that the range of the function is between 3 inclusive and 5 not inclusive. And now we can conclude our findings. So the domain is X belongs to. Negative infinity up to positive infinity and range is Y belongs to. 3 inclusive, up to 5 not inclusive, and those will be our final answers to this problem. Thank you for watching.

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

𝔂 = 2 + 3x² .
x² + 4

Key Concepts

Domain of a Function

Range of a Function

Behavior of Rational Functions

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