Combining two functions point by point, and the one case that fails

0:00There are five ways to build a new function out of ones you already have. Add them. Subtract them. Scale one of them by a number. Multiply them. Divide one by the other. Not one of the five invents anything. Every one is arithmetic applied to numbers the two functions hand you. Here is the shape of all five at once. Choose an input. Put it into the first function and read off a number. Put the same input into the second and read off another. Combine those two numbers, and that is what the new function sends your input to.

0:40Nothing happens to the functions themselves. The work is done at the level of values, one input at a time. Four of these five always work. One does not, and by the end you will know exactly which inputs it loses and why. First, what the two functions have to share. They have to accept the same things. If the second function will not take an input the first accepts, there is no second number to combine.

1:12So both are declared on one common set of inputs. Hand the machinery two functions on different sets and it refuses. Now what they do not have to share: their values. Take the squaring rule and the rule that doubles the input and adds one. Over a sweep of twenty-five inputs they disagree at every single one, because the place they would agree is not a number you can write as a fraction.

1:41And they combine without difficulty. Agreement is not what is asked for. A shared set of inputs is. Start with the two easiest. The sum sends each input to the sum of the two values there. The difference sends it to their difference. That is one definition written twice with the sign changed. Across the whole sweep the sum's value is the first value plus the second at every input, with zero exceptions, and the difference is the same two values with the other sign.

2:18Because each input is handled on its own, the order you walk them in cannot matter. Walk the twenty-five backwards and you get the same function: zero disagreements. Swapping the two functions is a different matter. The sum is untouched. The difference changes at all twenty-five inputs, and there is no input where it survives the swap. The third construction is the odd one in a different way. It does not take two functions at all.

2:52It takes one function and one number, and the number is called a scalar. The scalar multiplies the value the function produces. Be slow here, because there is a second thing it could have meant: multiply the input first, then apply the function. Those are different constructions, and most examples will not show you the difference. Take the identity. Scale its output by five, or its input by five, and you get the same function — zero disagreements across all twenty-five inputs. The rule that doubles: same story.

3:32Now the constant function that sends everything to three. Scale its output by five and every value becomes fifteen. Scale its input and nothing changes at all, because the function was never looking at the input. Those two readings disagree at every single input. Of four rules tested this way, two expose the difference and two cannot. The scalar meets the output, after the function has done its work. The fourth is the product, and it comes with a word attached: pointwise.

4:10The product sends each input to the two values multiplied together. Pointwise means what it says. At this input, these two numbers, multiplied here. Nothing is multiplied at the level of functions. Squaring against twice-plus-one. At minus two the values are four and minus three, so the product is minus twelve. At nought, nought. At one, three. At three, sixty-three. Collect those and the product is twice the input cubed plus the input squared, and that expression agrees with the pointwise product at every input in the sweep.

4:53The fifth is the quotient, and it is built by exactly the same manoeuvre. At each input, take the first value and divide it by the second. Same walk, one input at a time, nothing new invented. And this is the first construction that arrives carrying a condition: the second value must not be nought. Run the quotient of the square over twice-plus-one across the sweep and it drops exactly one input. Not a stretch, not a region — a single number, minus a half.

5:29The other three dropped nothing at all. Sum, difference, product — every one kept all twenty-five inputs. Four are unconditional and one is not, and the recipe was identical. The answer is not about functions. It is about numbers. Take the twenty-five values in the sweep. Can you add something to each? All twenty-five: yes. Multiply each by something? All twenty-five: yes. Is there something that multiplies it to give one? Over a search pool of one thousand one hundred and eleven fractions, twenty-four of the twenty-five find a partner and one does not.

6:15The one that does not is nought. Nothing multiplied by nought is one, so nought has no reciprocal, and no amount of searching will produce one. Adding, subtracting and multiplying are things the real numbers always permit. Dividing is not, and that single fact is the whole difference between the fifth construction and the other four. It also tells you where to look. The condition is on the second value, so you lose the inputs where the lower function is sent to nought.

6:51The lower one. Not the upper one, and this is where people go wrong. Here the upper rule, the square, is sent to nought at the input nought — and nought is not excluded. The quotient is happy there: nought over one is nought. The excluded input came from the lower rule, sent to nought at minus a half. That one, and only that one, had to go. All four combinations of that pair, properly.

7:21The sum: the square plus twice the input plus one. At minus two, one. At nought, one. At one, four. At three, sixteen. The difference: the square minus twice the input minus one. At the same four inputs: seven, minus one, minus two, two. The product: twice the input cubed plus the input squared. The quotient: the square over twice-the-input-plus-one, everywhere except minus a half. Now something the definitions never promised. That sum factorises: it is the square of one more than the input.

8:02Not a guess. Hunting over every pair of whole numbers from minus twelve to twelve, exactly one factorisation agrees with the sum at every input. Run the same hunt on the difference and it comes back empty — zero factorisations. Same recipe, same pair of rules, and one result is tidy while the other is not. The construction did not care. So finding the excluded inputs is one question asked over and over. Where is the lower function sent to nought?

8:40Twice the input plus one is nought at one input. Take a quadratic instead: the input squared, minus five times the input, plus four. It factorises into the input minus one and the input minus four, so it is nought at one and four. Another: the input squared, minus eight times the input, plus twelve, which is the input minus two against the input minus six. Zeros at two and six.

9:11And a linear one: twice the input minus three is nought at three halves. Put each of those underneath the square and count what the quotient loses. One, two, two, one. The number of inputs a quotient drops is the number where its lower function is sent to nought. Every time. So the whole skill is solving one equation. Set the lower rule to nought and find out where. A second pair, chosen because it makes the tidying interesting.

9:45The first rule takes the square root of its input. The second hands the input straight back. Both are declared on inputs that are not negative, because the first has nothing to give a negative number. We run them over eleven inputs where the root is exact — the perfect squares from nought to a hundred — so nothing here is a rounded decimal. The sum: the root plus the input. At nought, nought. At four, six. At nine, twelve. The difference: at the same three, nought, minus two, minus six.

10:23The product: the root times the input. Tidy it with the index laws and it is the input raised to three halves. Which you can check rather than accept. The root cubed and the pointwise product agree at all eleven inputs. At four the product is eight, and two cubed is eight. The quotient: the root over the input, which tidies to the input raised to minus one half. It drops one input — nought — because nought is where the lower rule, the identity, is sent to nought. At four it is a half, and it keeps ten of the eleven.

11:05Now the part worth slowing down for. There is a story students tell themselves: once you tidy the expression, the exclusion goes away. The root over the input breaks at nought — but it is really the input to the minus one half, one clean power, so surely it is fine. It is not fine. Ask the tidied form for its value at nought and it refuses, for the same reason and at the same input.

11:35Checked across all eleven inputs: the tidied form agrees with the quotient wherever both answer, and refuses exactly the input the quotient dropped. The excluded input belongs to the function, not to the way you wrote it down. And there are two separate restrictions in play here. The first: negative inputs were never declared, so they went before any combining started. The second: nought was declared, and the quotient lost it anyway. Nought is in the sum, the difference and the product. Of the four combinations, exactly one loses it.

12:16One restriction is a decision about what the functions accept. The other is a consequence of what the lower function does. One more thing the product is not. It is not feeding one function into the other. Multiplying two values at the same input, and putting one function's output into the other, are different operations. Only the first is one of the five. At the input two the square is four and twice-plus-one is five, so the pointwise product is twenty. Feed twice-plus-one into the square instead: five, squared, is twenty-five.

13:00Across the sweep those two agree at exactly one input and disagree at the other twenty-four. And that one input is minus a half — the very input the quotient had to drop. No coincidence. At minus a half the lower rule is nought, so the product is something times nought and the feeding is nought squared. They agree for a reason that has nothing to do with the operations being alike.

13:29Feed them the other way round and you get a third function again, and there is no input where all three agree. Three functions from the same two rules. The product is one of the five. The other two are not on the list. So what do the five share? Every one is defined at the level of values. Choose an input, read off what comes back, combine those numbers there.

13:58Four take two functions on one common set of inputs. The scalar multiple takes one function and a number, and the number meets the output. The scalar is a number, not a letter. Call it by another name and it is the same function; change the number and it is a different one, disagreeing at twenty-four of the twenty-five inputs. And exactly one of the four can lose an input.

14:26Over five different pairs of rules, sum, difference and product dropped nothing whatsoever. The quotient dropped something every time: one, one, two, two, one. It is one fact about the real numbers, arriving one input at a time: nought has no reciprocal. So when you meet a quotient, the question is never whether the expression looks safe. It is where the lower function is sent to nought. And write what you find beside the rule, because it is part of the function and it does not go away when the algebra gets tidier.