Algebra, Year 11: proof, inverse and composite functions

(2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1), which is a multiple of 2

HintWrite each odd number as one more than an even number, using a different letter for each.

The whyn and m are any whole numbers, so 2n + 1 and 2m + 1 stand for any two odd numbers. Taking out the factor 2 shows the total is 2 × (a whole number), which is the definition of even. Examples such as 3 + 5 = 8 illustrate it but do not prove it.

(n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1, which is odd

HintCall the numbers n and n + 1, square the larger and subtract the square of the smaller.

The why2n is even for every whole number n, so 2n + 1 is one more than an even number, which is odd. For example 7² − 6² = 49 − 36 = 13. The result 2n + 1 is also the sum of the two numbers, n + (n + 1).

2n × (2n + 2) = 4n² + 4n = 4(n² + n), which is a multiple of 4

HintWrite the smaller as twice a whole number; the next one is 2 more.

The whyConsecutive even numbers differ by 2, so they are 2n and 2n + 2, with the same n because the two numbers are linked. Taking out the factor 4 leaves n² + n, a whole number, so the product is 4 times a whole number. Stopping at 4n² + 4n, without the factorised line and a conclusion, leaves the proof unfinished.

Using 2n + 1 twice would make them the same number, so the proof would not cover two different odd numbers

HintPut n = 4 into each expression and see which pair of values you get.

The whyWith one letter, both expressions always have the same value (9 and 9 when n = 4), so the proof would only be about doubling an odd number. Two independent letters let the numbers be 9 and 23, or any other pair. For consecutive odd numbers, 2n + 1 and 2n + 3 is right, because then they are linked.

A counter-example

HintIts name starts with a prefix meaning 'against'.

The whyThe statement 'all prime numbers are odd' is disproved by the number 2. One case that fails is enough to disprove a statement, whereas no number of cases that work is enough to prove one.

The inverse function of f, written f⁻¹

HintIts name is the word used for operations that undo each other, such as adding and subtracting.

The whyIf f(x) = 2x + 3 turns 5 into 13, then f⁻¹ turns 13 back into 5. It undoes each operation of f in reverse order: subtract 3, then halve, so f⁻¹(x) = (x − 3)/2.

f⁻¹(x) = (3x − 1)/2

HintWrite y = (2x + 1)/3 and rearrange to make x the subject.

The whyy = (2x + 1)/3 gives 3y = 2x + 1, then 3y − 1 = 2x, so x = (3y − 1)/2. Writing the answer with x as the input gives f⁻¹(x) = (3x − 1)/2. Check: f(4) = 3 and f⁻¹(3) = 4.

4

HintAsk which input the function would turn into 15.

The whyf⁻¹(15) is the input that gives the output 15, so solve 2x + 7 = 15: x = 4. Or use f⁻¹(x) = (x − 7)/2. Check: f(4) = 15.

No: f⁻¹(x) = x/2, which undoes the doubling, but 1/f(x) = 1/(2x)

HintTry the input 6 in each one.

The whyf⁻¹(6) = 3, because f(3) = 6, whereas 1/f(6) = 1/12. The small raised −1 after a function's name means 'inverse function', not 'reciprocal'.

x

HintOne function undoes whatever the first one did.

The whyApplying f and then f⁻¹ returns you to where you started. With f(x) = 3x − 5 and f⁻¹(x) = (x + 5)/3: f(4) = 7 and f⁻¹(7) = 4. This gives a quick check on any inverse you find.

A composite function

HintIts name means 'made up of several parts'.

The whyfg(x) means 'do g first, then do f to the answer'. If g doubles and f adds 3, then fg(x) = 2x + 3. It can always be written as one single rule.

7

HintStart with the function written nearest to the 2.

The whyg acts first: g(2) = 4. Then f(4) = 4 + 3 = 7. The function next to the bracket is always applied first.

4x² + 4x + 1

HintPut the whole of f(x) into g in place of x.

The whygf(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1. f is applied first because it is nearest the x, and then g squares the whole result.

g

HintRead outwards from the x.

The whyfg(x) is short for f(g(x)). The x goes into g, and the output of g goes into f. So the functions are applied from right to left, the opposite of the reading order.

No: fg(x) = 2x + 3, but gf(x) = 2(x + 3) = 2x + 6

HintWork each one out as a single rule and compare.

The whyfg doubles and then adds 3; gf adds 3 and then doubles, so the 3 gets doubled as well. The order in which functions are applied usually changes the result.