Algebraic Combinations

In Ex...

Question

Algebraic Combinations

In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.

f(x) = √(x + 1), g(x) = √(x − 1)

Accepted Answer

Welcome back everyone to another video. Given the functions U of X equals square root of 5 minus X and V of X equals square root of X + 2, determine the domains of U, V, U + V and U multiplied by V. So let's begin with the first domain. We want to identify the domain of U. Now U of X. is a function which is square root of 5 minus x, and we have to recall that we cannot take square root of a negative number in a real plane, right? So essentially we want to make sure that 5 minus X is greater than or equal to 0. And now if we move negative X to the right hand side. We get 5 is greater than or equal to X, or simply speaking, X is less than or equal to 5. In an interval notation, we can say that X belongs to the interval from negative infinity up to 5. So that's the domain for you. Now for V, we know that our function is square root of X + 2. We're going to use the same logic. X + 2 must be greater than or equal to 0, which means that X is greater than or equal 2. -2 if we subtract 2 from both sides, right, so in an interval notation X belongs to the interval from negative to inclusive up to infinity, and that would be our second answer. Now what is U + V of X? Well, we're getting a new function which is a composition of the two functions. So this gives us square root of. 5 minus x plus square root of X + 2. And now we have to be very careful. Our new function has two radicals, right? So essentially we want to satisfy two conditions. We want to make sure that 5 minus X is greater than or equal to 0 and X + 2 is greater than or equal to 0. So in this case, we're looking for the intersection of the two previous domains. We can plot a number, a line for a complete visualization, and we're going to add our critical points, so those would be -2 and 5, right? So -2 and 5. So, for the first solution, we had an interval from negative infinity up to 5, right? So, essentially from negative infinity up to 5. For the second one, we had from negative to up to infinity, and now we're looking for the intersection. So what we can see is that The common domain would be from -2 to 5, which satisfies them both, right? So in this case, X belongs to the interval from -2 up to 5 inclusive. And now for the product, you multiplied by B of X. The main difference is that we are now multiplying the two functions, square root of 5 minus X, multiplied by square root of X plus 2. And once again, we want to make sure that we're satisfying each, so the domain will be the same, right? It doesn't matter if we're adding them or multiplying them. We want to make sure that each Radical has a function that is greater than or equal to zero within the radical. So in this case, X belongs to the interval from -2 to 5 as well, and those will be our answers. So let's label them. For ux belongs from negative infinity up to 5 inclusive or V from negative to inclusive up to infinity. Then we have U + V. And you multiplied by V. Those two domains would be from negative to inclusive, up to 5 inclusive. Thank you for watching.

Algebraic Combinations

In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.

f(x) = √(x + 1), g(x) = √(x − 1)

Key Concepts

Domain of a Function

Algebraic Combinations of Functions

Square Root Function Properties

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