Find 𝑓 minus 𝑔 of 𝑥 given 𝑓 of 𝑥 equals 𝑥 squared minus two and 𝑔 of 𝑥 equals 𝑥 minus five.
𝑓 minus 𝑔 of 𝑥 is a combination of functions. In fact, given two functions 𝑓 and 𝑔, 𝑓 minus 𝑔 of 𝑥 is simply 𝑓 of 𝑥 minus 𝑔 of 𝑥. Now, in this case, 𝑓 of 𝑥 is the function 𝑥 squared minus two, whilst 𝑔 of 𝑥 is 𝑥 minus five. So we can say that 𝑓 minus 𝑔 of 𝑥 must simply be 𝑥 squared minus two minus 𝑥 minus five. Now, of course, we do need to be really careful here when subtracting 𝑥 minus five. This is the same as subtracting each part of the expression inside our parentheses. So we’re subtracting 𝑥 and we’re subtracting negative five. This is the same as subtracting 𝑥 and adding positive five. So we can write 𝑓 minus 𝑔 of 𝑥 as 𝑥 squared minus two minus 𝑥 plus five.
Then, to simplify this expression, we’ll collect like terms. Negative two plus five is positive three. So this simplifies to 𝑥 squared minus 𝑥 plus three. And so given 𝑓 of 𝑥 and 𝑔 of 𝑥 in this case, 𝑓 minus 𝑔 of 𝑥 is 𝑥 squared minus 𝑥 plus three.