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

Question

𝔂 = 2e⁻ˣ - 3

Accepted Answer

Welcome back, everyone. Find the domain and range of the function G of X equals -3 eats the power of x + 4. So first of all, for this problem, let's understand that we are dealing with an exponential function and specifically what we're going to do is we call that an exponential function in the form of a. So the power of X, let's suppose that we have Y equals 8 is the power of X, in this case G of X. Or why is equal to. 3 it's the power of X + 4 we can ignore 4 and -3 for now and just focus on e to the power of x. Well, this exponential function is essentially defined for all the X values, right? Essentially. Our exponential function keeps increasing. We can say that this represents a curve E to the power of X. And essentially we can say that the domain of the given function as well will be all real numbers because we define the domain as a set of x values for which the function is defined. We can raise our base of E to any number we want, and the function will still be defined. Between negative infinity and infinity, so we can say that the domain of the given function is X belongs to all real numbers or from negative infinity up to infinity, right? There is no point that we cannot use for X that does not allow us to evaluate the value of the function. Now what about the range? Well, first of all, we have to recall that if we begin with eat to the power of X, its range is all positive values, right? So it's the power of X is greater than 0, right? Essentially when X goes towards negative infinity, it's approaching 0 but never gets to 0. When X gets really large, eats the power of X goes to infinity. So what we're going to do is simply manipulate this function. And specifically we're going to focus on the inequality, and the first step is to multiply both sides by -3. We are performing this because we have -3 as the leading coefficient, so we're going to get -3 E to the power of X is going to be less than 0. Notice that we are flipping the sign because we're multiplying by a negative number, and now what we're going to do is simply add 4 to both sides. So 38, the power of X + 4 must be less than 4. And that's it. This is how we have successfully obtained the range of the function. By using the parent function e to the power of X, so the range of G of X is Y belongs to the interval from negative infinity. Up to 4 because it says it must be less than 4, right, so the upper bound is 4 and the lower bound is negative infinity. That's it. We have got our answers and thank you for watching.

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

𝔂 = 2e⁻ˣ - 3

Key Concepts

Domain of a Function

Range of a Function

Exponential Functions

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