0580 · 2.13Functions

Function notation and evaluation

0:00Imagine a machine that multiplies your number by three, then subtracts five. An input of two produces an output of one. Functions describe this relationship between inputs and outputs. By the end, you will understand permitted inputs, possible outputs, reversing a function, and connecting two functions together. You need substitution, the order of operations, and simple equation rearrangement. Substitution means replacing a letter with a given number wherever that letter appears.

0:27Brackets keep an expression together so that the correct operations happen first. We will use these familiar skills to explain what function notation actually means. A function assigns exactly one output to each input in its domain. Think of the domain as the collection of permitted inputs. Once you choose an allowed input, the function must give one definite answer. Different inputs can produce the same output, but one input cannot produce two different outputs.

0:53We can name our machine f and write its rule using function notation. We say f(x) = 3x − 5. The letter f names the function, while x represents its input. The expression 3x − 5 tells us how to calculate the output from that input. The notation f of x does not mean f × x. It means the output of function f when its input is x. Likewise, f of two means the output when the input is two.

1:19The value inside the function brackets tells you what to substitute into the rule. For our first worked example, find f of two. Replace x with two to obtain three multiplied by two minus five. Multiplication comes first, giving 6 − 5. Subtracting five gives one, so f of 2 = 1. That answer is an output, not a new rule for the whole function. The rule remains 3x − 5, regardless of which permitted input we choose.

1:45If a question instead gives the output and asks for the input, we solve an equation. For example, an output of one means 3x − 5 = 1.

Domain, range and restrictions

1:54A mapping diagram shows the relationship using arrows instead of a formula alone. Each permitted input appears on the left, and its output appears on the right. Every input must have exactly one outgoing arrow. Two arrows may arrive at the same output, because different inputs can share an answer. Here, h(x) = 2x² + 3. The chosen domain contains negative two, zero and two. Both negative two and positive two produce eleven, while zero produces three.

2:22The range is the collection of outputs actually produced, so this diagram has range three and eleven. For negative two, substitute carefully to obtain 2 × −2², plus three. Squaring negative two gives four, leaving two multiplied by four plus three. The multiplication gives eight, and adding three produces eleven. Substituting zero gives three, while substituting positive two also gives eleven. Keep brackets around a negative input when writing your substitution.

2:51The square applies to the whole input, including its negative sign. −2² is positive four because two negative factors multiply to give a positive result. Losing those brackets can change the calculation and therefore the output. The domain matters because it controls which outputs are possible. If h allows every real number as an input, its range is all numbers greater than or equal to three. A square cannot be negative, so twice a square cannot be negative either.

3:17Adding three makes three the smallest possible output, achieved when x is zero. For f(x) = 3x − 5, every real input is permitted unless a question gives a restriction. Its range is also all real numbers, because any chosen output can be reached. Given an output, add five and divide by three to obtain its input. Domain and range describe the input side and output side respectively. Some rules have an input restriction built into them.

3:43Consider r of x = 3 divided by the quantity x + 2. An input of negative two makes the denominator zero, and division by zero is undefined. Its domain therefore contains every real number except negative two. This function cannot produce zero, because its numerator is three rather than zero. Every other real output is possible by choosing a suitable denominator. Its range is therefore every real number except zero.

4:08Notice that the excluded input and excluded output are different numbers, because domain and range describe different parts of the relationship.

Inverse functions and restrictions

4:16An inverse function reverses the original function. It takes an original output and returns the input that produced it. Our function first multiplies by three and then subtracts five. To undo the complete process, we add five first and then divide by three. The order reverses because the last operation must be undone first. Think of putting on socks and then shoes. To reverse that process, remove the shoes before removing the socks.

4:41In the same way, reversing our function starts by undoing the subtraction rather than the multiplication. The inverse notation is read as f inverse of x. Although its written symbol contains negative one, it does not mean 1 ÷ f(x). A reciprocal changes a value by dividing one by it. An inverse instead reverses the relationship between the original inputs and outputs. To find the inverse algebraically, first write y = 3x − 5.

5:08Swap x and y to obtain x = 3y − 5. This swap makes the old output the new input. We now rearrange the equation to express the new output y in terms of x. Add five to both sides, giving x + 5 = 3y. Divide both sides by three to obtain y equals the quantity x plus five divided by three. Therefore, f inverse of x equals the quantity x plus five divided by three.

5:31This agrees with our reverse machine: add five, then divide by three. Check the result using our earlier input and output. We found that f takes two to one. Its inverse takes one to the quantity one plus five divided by three. That becomes 6 ÷ 3, giving two and recovering the original input. Now find the inverse of g(x) = 3 times the quantity x + 4, all divided by five.

5:56Swapping x and y gives x = 3 times the quantity y + 4, over five. Multiplying both sides by five and expanding gives 5x = 3y + 12. Subtracting twelve and dividing by three gives g inverse of x equals the quantity 5x − 12, all over three. An inverse function is only possible when each original output identifies one original input. Our earlier squaring example sent both negative two and positive two to eleven.

6:24Reversing those arrows would give two answers for an input of eleven. That reverse relationship would not be a function, because one input would have two outputs. A suitable domain restriction can remove that ambiguity. For example, allowing only inputs of zero or more for h means that eleven comes from positive two, not negative two. The original range becomes the inverse domain, and the original domain becomes the inverse range.

6:47Reversing the arrows explains this exchange without needing a separate rule to memorise.

Composite functions and exam practice

6:50A composite function connects two functions in sequence. The output from the first becomes the input to the second. Cambridge defines gf(x) as g of f of x. This means apply f first, then apply g to the result. The function nearest x acts first. Similarly, fg(x) means f of g of x, so g acts before f. These expressions do not mean multiplying two function values together. They describe feeding one complete output into another rule.

7:20For our third worked example, use f(x) = 3x − 5. Let g of x equal three times the quantity x + 4, divided by five. We are asked to find g of f of x. Begin with f because its output becomes the input to g. The rule for g adds four to its input, multiplies by three, then divides by five. Replace its input with the whole expression 3x − 5.

7:44Inside the brackets, subtracting five and adding four simplifies to subtracting one. Expanding the numerator gives 9x − 3, all divided by five. Reversing the order means placing the complete expression for g into f. This gives 3 × g(x), then minus five. 3 × 3 makes nine, so we have nine times the quantity x + 4, all over five, then minus five. Writing five as 25/5 and combining the numerators gives 9x + 11, all divided by five.

8:14The two orders produced different expressions, so function order matters. Adding something before multiplying usually differs from adding it afterwards. Read the notation before substituting, and keep the complete substituted expression inside brackets. You are not expected to find the domains and ranges of composite functions in this syllabus. Here is an exam-style question using a new pair of functions. Let f of x equal three divided by the quantity x + 2.

8:40Let g of x equal the square of the quantity 3x + 5. Find f of g of x, giving your answer as a fraction in its simplest form. Apply g first, then substitute its entire output into the denominator of f. This gives three, divided by the quantity 3x + 5, all squared, plus two. Expanding the square gives 9x² + 15x + 15x + 25. The additional two remains in the denominator.

9:05Combining terms gives three over the quantity 9x² + 30x + 27. Factor three from the whole denominator to obtain three times the quantity 3x² + 10x + 9. Cancel the common factor of three between numerator and denominator. The simplest fraction is one over the quantity 3x² + 10x + 9. A common mistake is cancelling three against just one term in the denominator. Cancellation works with factors of the entire numerator and denominator, not isolated terms in a sum.

9:36Another mistake is applying the functions in the wrong order. Writing the nested function notation first makes the required sequence clear. These substitutions, inverse rearrangements and composite simplifications must work without a calculator on Paper two. They may also appear on calculator Paper four, but the algebraic reasoning is unchanged. Keep answers exact when a fraction or algebraic expression is requested. For numerical answers requiring rounding, normally use three significant figures unless instructed otherwise, and avoid rounding during intermediate calculations.

10:08A function gives exactly one output for each permitted input. Its domain contains the allowed inputs, and its range contains the outputs actually reached. An inverse reverses that relationship when each output identifies a unique original input. A composite joins two rules, with the function nearest the input acting first.