Video: Finding the Difference between Two Functions Then Determining Its Domain

If 𝑓₁ : (βˆ’7, 8] ⟢ ℝ where 𝑓₁(π‘₯) = π‘₯ βˆ’ 2, and 𝑓₂ : [βˆ’8, 4] ⟢ ℝ where 𝑓₂(π‘₯) = 4π‘₯Β² + 8π‘₯ + 3, find (𝑓₂ βˆ’ 𝑓₁)(π‘₯) and the (𝑓₂ βˆ’ 𝑓₁).

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Video Transcript

If 𝑓 one maps numbers from the left-open right-closed interval from negative seven to eight onto the set of real numbers, where 𝑓 sub one of π‘₯ is equal to π‘₯ minus two, and 𝑓 sub two maps numbers from the closed interval negative eight to four onto the set of real numbers, where 𝑓 sub two of π‘₯ is equal to four π‘₯ squared plus eight π‘₯ plus three, find the function 𝑓 two minus 𝑓 one of π‘₯ and the domain of 𝑓 two minus 𝑓 one.

Let’s begin by recalling what we mean by this combination function, 𝑓 two minus 𝑓 one of π‘₯. We know that the combination function 𝑓 minus 𝑔 of π‘₯ is simply the difference between these two functions. It’s 𝑓 of π‘₯ minus 𝑔 of π‘₯. And so in this case, 𝑓 two minus 𝑓 sub one of π‘₯ must be the function 𝑓 sub two of π‘₯ minus the function 𝑓 sub one of π‘₯. We see that 𝑓 sub two of π‘₯ is this quadratic four π‘₯ squared plus eight π‘₯ plus three and 𝑓 sub one of π‘₯ is π‘₯ minus two. So 𝑓 sub two minus 𝑓 sub one of π‘₯ is four π‘₯ squared plus eight π‘₯ plus three minus π‘₯ minus two.

Now, if we distribute the parentheses, we get negative π‘₯ minus negative two as our last two terms. And of course, minus negative two is just two. So we have four π‘₯ squared minus eight π‘₯ plus three minus π‘₯ plus two, which simplifies to four π‘₯ squared minus nine π‘₯ plus five. And so we found the function 𝑓 two minus 𝑓 sub one of π‘₯. But what is its domain? Well, when we combine functions, we know that the domain is the intersection of the domains of the functions we started with. And so we’re looking to find the intersection or the overlap of the domains of 𝑓 sub one and 𝑓 sub two.

Now, of course, 𝑓 sub one of π‘₯ and 𝑓 sub two of π‘₯ are polynomials. And usually, we say that the domain of a polynomial is a set of real numbers. But actually, we’re told that 𝑓 sub one maps numbers from the left-open right-closed interval from negative seven to eight. So this is the domain of 𝑓 sub one. Similarly, we’re told that 𝑓 sub two maps numbers from the closed interval negative eight to four. And so that’s the domain of 𝑓 sub two. And this means the domain of 𝑓 sub two minus 𝑓 sub one is the overlap of these two domains. We want to find the values of π‘₯ that satisfy both.

And so to help us visualize this, we’re going to use a number line. Let’s start off by mapping the domain of 𝑓 sub one. It takes values of π‘₯ greater than negative seven and less than or equal to eight. So the domain of 𝑓 sub one is shown. 𝑓 sub two takes values of π‘₯ greater than or equal to negative eight and less than or equal to four. So the pink line represents the domain 𝑓 sub two.

We’re interested, of course, in the overlap of these. And so we’re going to be beginning at negative seven; we’re going to be ending at four. We’re not going to include the value of negative seven, but we are going to include the value of four. And so we can represent this using interval notation as shown. The domain of 𝑓 sub two minus 𝑓 sub one is the left-open right-closed interval from negative seven to four.

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