Functions flashcards
9 cards on Functions: the key points, worked examples and the slips to watch. Swipe through them, or print them.
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Functions
Function notation
What does f(x) mean, and how do you evaluate a function for a number?
Answer it in your head, then turn the card over.
Function notation
Key points
- A function is a rule that turns each input into exactly one output.
- f(x) is read "f of x": the output when x goes in. f(4) is the output when 4 goes in.
- To evaluate, replace every x with the number, then work it out in the usual order.
f(x) = 3x − 2
f(4) = 3 × 4 − 2 = 10
f(−1) = 3 × (−1) − 2 = −5
Watch out
Put a negative input in brackets before squaring: for f(x) = x² + 1, f(−3) = 9 + 1 = 10, not −8.
Functions
An expression as the input
How do you find f(2a) or f(x + 1) when you know f(x)?
Answer it in your head, then turn the card over.
An expression as the input
Key points
- Replace each x with the whole expression, in brackets.
- Everything else in the rule stays as it is.
- Expand and simplify, then test your answer with a number.
f(x) = 4x − 1
f(2a) = 4(2a) − 1 = 8a − 1
Test with a = 3: f(6) = 4 × 6 − 1 = 23
and 8 × 3 − 1 = 23, so it matches.
Watch out
f(2a) is not 2 × f(a): 2f(a) = 8a − 2, but f(2a) = 8a − 1.
Functions
Solving f(x) = k
How do you solve an equation like f(x) = 17?
Answer it in your head, then turn the card over.
Solving f(x) = k
Key points
- It asks which input gives an output of 17: the reverse of evaluating.
- Write the rule equal to 17, then solve it as an ordinary equation.
- Check by putting your answer back into f.
f(x) = 2x + 5, solve f(x) = 17
2x + 5 = 17
2x = 12, so x = 6
Check: f(6) = 2 × 6 + 5 = 17
Watch out
Working out f(17) = 39 answers a different question: f(x) = 17 wants the input, not the output.
Functions
When f(x) = k has two answers
Why can an equation like x² − 4 = 21 give two values of x?
Answer it in your head, then turn the card over.
When f(x) = k has two answers
Key points
- With an x² in the rule, two different inputs can give the same output.
- A positive number has two square roots, one positive and one negative.
- Give both, written as x = 5 or x = −5.
f(x) = x² − 4, solve f(x) = 21
x² − 4 = 21
x² = 25
x = 5 or x = −5
Watch out
Stopping at x = 5 loses the second answer: (−5)² is 25 as well.
Functions
Composite functions
What does fg(x) mean, and how do you find it as an expression?
Answer it in your head, then turn the card over.
Composite functions
Key points
- fg(x) means do g first, then put the result into f.
- Work from the letter nearest the x outwards.
- Replace the x in f with the whole of g(x), in brackets, then simplify.
f(x) = x + 4 and g(x) = 3x
fg(x) = f(3x) = 3x + 4
gf(x) = g(x + 4) = 3(x + 4) = 3x + 12
Test with x = 2: fg(2) = 10 and gf(2) = 18
Watch out
Doing f first because it is written first gives gf(x), not fg(x).
Functions
A composite function of a number
How do you work out fg(2) and gf(2), and why are they usually different?
Answer it in your head, then turn the card over.
A composite function of a number
Key points
- Work from the inside out: find g(2) first, then put that answer into f.
- gf(2) is the other way round: f(2) first, then into g.
- Order matters: fg and gf are rarely equal. ff(x) means do f twice.
f(x) = x² and g(x) = x + 3
fg(2): g(2) = 5, then f(5) = 5² = 25
gf(2): f(2) = 4, then g(4) = 4 + 3 = 7
Watch out
fg(x) is not f(x) × g(x): it is one function fed into the other.
Functions
Finding an inverse function
What are the steps for finding f⁻¹(x)?
Answer it in your head, then turn the card over.
Finding an inverse function
Key points
- Write y = f(x).
- Swap x and y.
- Rearrange to make y the subject. That expression is f⁻¹(x).
f(x) = 2x + 6
y = 2x + 6, swap: x = 2y + 6
x − 6 = 2y, so f⁻¹(x) =
Check: f(1) = 8, and = 1
Watch out
Undo the steps in reverse order: the inverse of 3x − 2 is , not + 2.
Functions
What an inverse does
What does f⁻¹(x) do, and how can you check the inverse you found?
Answer it in your head, then turn the card over.
What an inverse does
Key points
- f⁻¹ undoes f: it takes each output back to the input it came from.
- So if f(4) = 10, then f⁻¹(10) = 4. Doing f and then f⁻¹ gets you back where you started.
- To check: put a number into f, then put the output into your f⁻¹(x).
f(x) = 3x − 2 and f⁻¹(x) =
f(4) = 3 × 4 − 2 = 10
f⁻¹(10) = = = 4, back to the start
Watch out
The ⁻¹ means undo, not one over: for f(x) = x + 5, f⁻¹(x) = x − 5, not .
Functions
The inverse of a fraction
How do you find the inverse when f(x) is written as a fraction?
Answer it in your head, then turn the card over.
The inverse of a fraction
Key points
- Swap x and y as usual.
- Multiply both sides by the denominator to clear the fraction.
- Then undo the rest, one step at a time, until y is on its own.
f(x) =
Swap: x =
× 5: 5x = 2y + 1
So 2y = 5x − 1 and f⁻¹(x) =
Check: f(2) = = 1, and = 2
Watch out
Keep the brackets: halves all of 5x − 1, but 5x − halves only the 1.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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