Inverse and Composite Functions

f g of x equals twenty-three. f is a machine that adds three, g is a machine that doubles, and you need the x that went in. Those two letters side by side are not a multiplication. They are an order, one machine and then the other, and the order changes the answer. To get back to x, you undo both machines. And to undo them in the right order, you first need to know which one ran first.

This is video two of six in Functions and Differentiation, and it is Higher material. If reversing a single machine feels shaky, the video Function Machines: Inputs, Outputs and Reversing covers that step.

A function is a rule that turns each input into exactly one output. A function machine is that rule drawn as a box, with the input going in one side. A composite function is two of those machines joined end to end. The output of the first machine becomes the input of the second. It is written with the two letters side by side, so f g of five means five goes through both f and g, one after the other. Those two letters can be read in more than one way, and only one reading is right. f adds three and g doubles. Is f g of five eighty, sixteen or thirteen? Thirteen. The five goes into g first, because g is the machine sitting beside the five. g doubles five, and five doubled is ten. That ten goes into f, which adds three, and ten plus three is thirteen. Now, where did eighty go wrong, and where did sixteen go wrong? Eighty multiplied the two outputs, f of five times g of five, which is eight times ten. But f g is not a product, so multiplying is the wrong operation. Sixteen ran f first: five plus three is eight, and eight doubled is sixteen. That is g f of five, a different composite with the letters swapped.

Written out in full, f g of x is f of, open bracket, g of x, close bracket. With that bracket in place, which machine runs first, and why? g runs first, because g of x is inside the bracket. In any calculation the inside of a bracket comes before the outside, so g of x is found and then handed to f. One examiner's report on a Higher paper says this about composites. In weaker responses g h of x and or h g of x were often interpreted as multiplying g of x and h of x together. That is the eighty from the f g of five question, two outputs multiplied. The fix is to write the bracket out before any algebra, because once f g of x reads f of g of x, there is nothing to multiply. Now the same idea with x in place of a number. To find f g of x, start inside the bracket, where g of x is two x. Next, feed two x into f. f adds three to whatever goes in, so f g of x is two x plus three. Put five in as a check: two fives are ten, plus three is thirteen, the same thirteen the machines gave. Swap the letters and you get g f of x, which builds a different function with its own expression. Pause and write g f of x as an expression, then check it against mine. As an expression, g f of x is two x plus six. This time f is inside the bracket, so x goes into f and comes out as x plus three. Then g doubles all of that, and two lots of x plus three is two x plus six. f g of x is two x plus three, and g f of x is two x plus six. The same two machines in a different order make a different function.

An inverse function is the machine that undoes another one. Whatever the first machine did to a number, its inverse brings that number back. The inverse of f is written f to the minus one of x, and said out loud as f inverse of x. The raised minus one is a label for undo, not a power, so it never means one over f of x. Take h of x equals three x minus four. Going forwards, h multiplies by three, then subtracts four. Which of those two steps do you undo first, and why? The last step, because h did it last. Getting dressed works the same way: socks go on before shoes, but shoes come off first. The last step done is the first step undone. You undo each operation, and you undo them in reverse order. Now try it: find h to the minus one of x by undoing both steps. The inverse takes h's output in as its input, and calls it x. The subtract four went last, so undo it first: add four, and x becomes x plus four. Then undo the times three by dividing by three. All of x plus four goes over three. h to the minus one of x is x plus four, all over three. Keep all of x plus four on top of the fraction line, because the four is added before you divide. A correct inverse sends every output back to the input it came from. So test yours on a number you have not tried yet. What is h to the minus one of seventeen? Seven. Seventeen plus four is twenty-one, and twenty-one divided by three is seven. Run seven forwards through h: three sevens are twenty-one, take away four is seventeen, the number we started with. Another examiner's report, on a question about composites and inverses, puts it plainly. For many the notation alone was the stumbling block. In plain words, many students were stopped by reading the symbols, before any of the algebra had begun. The habit that fixes it is one line in words before the algebra: which machine runs first, which runs second, and which one you are undoing.

Now back to the question from the start: f g of x equals twenty-three, where f adds three and g doubles. Which machine do you undo first here, f or g, and why? f, because f ran last on the way in. Shoes come off before socks, so the last machine is the first one undone. Next, undo both machines in that order. What is x? x is ten. First undo f, which means taking three away, and twenty-three take away three is twenty. Undoing g means halving, and half of twenty is ten. So x equals ten. Check it forwards: g doubles ten to twenty, and f adds three to make twenty-three, the number the question gave. Put a finger on the x in f g of x. The letter nearest it, g, is the machine x goes into first, and it is the last one undone on the way back. The same holds inside one function. In three x minus four, the times three touches the x, so it goes in first and comes off last. Your handle for this video: nearest the x goes in first, and comes out last.

Let's check all of that with three quick questions, two of them on functions you haven't met. p takes away one, and q multiplies by five. What is p q of two? Nine. q sits nearest the two, so it goes first: two times five is ten, and ten take away one is nine. Next one. What does f to the minus one of x mean? The inverse of f, the machine that undoes f. The minus one is a label, never one over f of x. Here's a different one. k halves, then adds five. What does its inverse do first? It takes away five, because adding five was the last step in. Then it doubles, so k to the minus one of x is two lots of x minus five. And the twenty-three from the start came back to ten by undoing the order: nearest the x goes in first, and comes out last.

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Next in the chapter: Formal Function Notation: Domain and Range.

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