If 𝐹 one of 𝑥 and 𝐹 two of 𝑥 are both antiderivatives of the same function 𝑓 of 𝑥, what is the relation between 𝐹 one and 𝐹 two? (a) 𝐹 one minus 𝐹 two is a constant. (b) 𝐹 one must be the double of 𝐹 two. (c) 𝐹 one must be equal to 𝐹 two. (d) 𝐹 one plus 𝐹 two must be a constant. Or (e) only one of them is undefined.
We recall that the general antiderivative of a function lowercase 𝑓 of 𝑥 is the function capital 𝐹 of 𝑥 plus 𝐶 such that the first derivative of capital 𝐹 of 𝑥, 𝐹 prime of 𝑥, is equal to the function 𝑓 of 𝑥 and 𝐶 is any real constant. In other words, an antiderivative of a function lowercase 𝑓 of 𝑥 is another function capital 𝐹 of 𝑥, whose derivative is equal to the original function. An antiderivative is not unique. And there are many functions which differ up to a constant which give the same derivative.
We’re given that 𝐹 one and 𝐹 two of 𝑥 are both antiderivatives of the same function 𝐹 of 𝑥. So let’s suppose that 𝐹 one of 𝑥 is the function 𝐹 of 𝑥 plus 𝐶 one and 𝐹 two of 𝑥 is the function 𝐹 of 𝑥 plus 𝐶 two for two constants 𝐶 one and 𝐶 two. These functions differ only in the value of their constants. It follows then that if we subtract one function from the other, the result is 𝐶 one minus 𝐶 two, which is a constant. This is option (a) 𝐹 one minus 𝐹 two is a constant. Indeed, it’s also true that 𝐹 two minus 𝐹 one is a different constant. Let’s consider though the other four options we were offered.
Option (b) states that 𝐹 one must be the double of 𝐹 two. This would lead to 𝐹 of 𝑥 plus 𝐶 one is equal to two 𝐹 of 𝑥 plus two 𝐶 two, which may be true for certain values of 𝑥 in the domain of 𝐹, but it won’t be true for every single value. Option (c) states that 𝐹 one must be equal to 𝐹 two. But as we’ve seen, antiderivatives are not unique, and there are infinitely many possible antiderivatives for a given function, all differing by a constant.
For option (d), 𝐹 one plus 𝐹 two would be equal to two 𝐹 of 𝑥 plus 𝐶 one plus 𝐶 two or two 𝐹 of 𝑥 plus a constant 𝐶, which is not constant as the value of the function 𝐹 of 𝑥 will vary as 𝑥 varies. This would only be true if 𝐹 of 𝑥 is itself a constant function. But it isn’t true in general. For option (e), we’ve seen that both functions are defined, so neither is undefined, so we can rule this out too. Our answer is option (a). If 𝐹 one of 𝑥 and 𝐹 two of 𝑥 are both antiderivatives of the same function lowercase 𝑓 of 𝑥, then 𝐹 one minus 𝐹 two is a constant.