A function is a rule, made of one or more mathematical operations, that turns each input into an output . It can be thought of as a machine: numbers go in, the operations are applied, and new numbers come out.

Define a composite function .

A composite function applies one function to the output of another. The input goes through the first function, and the result then becomes the input of the second function.

Define the inverse function of a function.

The inverse function does the opposite of the original function, taking each output back to the input it came from. Putting a number through a function and then through its inverse gives back the number you started with.

GCSEMathsOCRHigherFlashcards6. AlgebraFunctions

Functions (OCR GCSE Maths: Higher): Flashcards

Exam code: J560

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FrontIntroduction to Functions
Define a function.

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Cards in this collection (16)

Define a function.
A function is a rule, made of one or more mathematical operations, that turns each input into an output.
It can be thought of as a machine: numbers go in, the operations are applied, and new numbers come out.
Write the function $x \to \times 5 \to + 6 \to y$ as an equation.
Apply the operations to $x$ in the order shown, which gives this equation:
$y = 5 x + 6$
The multiplication comes first, which is why it is $5 x + 6$ and not $5 (x + 6)$ .
Fill in the missing word:
A diagram in which each input is joined by an arrow to its output is called a $______$ diagram.
The completed sentence is:
A diagram in which each input is joined by an arrow to its output is called a mapping diagram.
Turning inputs into outputs like this is called mapping.
Write the equation $y = 2 (x + 3)$ as a function machine with input $x$ and output $y$ .
The bracket is worked out first, so the machine adds 3 and then multiplies by 2:
$x \to + 3 \to \times 2 \to y$
What is the output of $input \to \times 3 \to - 8 \to output$ when the input is $k$ ?
Put $k$ in as the input and apply each operation in turn, which gives an output of $3 k - 8$ .
An input that is a letter produces an output that is an algebraic expression rather than a number.
Define a composite function.
A composite function applies one function to the output of another.
The input goes through the first function, and the result then becomes the input of the second function.
Function A is $\times 3 \to - 8$ and function B is $+ 5 \to \times 2$ . What is the output when 7 is put through A and then B?
Function A gives $7 \times 3 - 8 = 13$ , and this output becomes the input to B.
Function B then gives $(13 + 5) \times 2 = 36$ , so the output is 36.
True or False?
Putting $x$ through the function $\times 2$ and then the function $+ 5$ gives a different output from putting it through $+ 5$ first and then $\times 2$ .
True.
The first order gives $2 x + 5$ but the second gives $2 (x + 5) = 2 x + 10$ , which is always 5 more.
So in a composite function the order in which the two functions are applied matters.
Function P is $\times 4$ and function Q is $- 3 \to \times 2$ . Find an expression for the output when $x$ goes through P and then Q.
Function P turns $x$ into $4 x$ , which then goes through Q to give $(4 x - 3) \times 2$ .
So the output is $8 x - 6$ .
Function A is $\times 3 \to - 8$ and function B is $+ 5 \to \times 2$ , and putting $x$ through A and then B gives an output of $4 x$ . Find the value of $x$ .
Putting $x$ through A gives $3 x - 8$ , and B then gives $(3 x - 8 + 5) \times 2 = 6 x - 6$ .
Setting $6 x - 6 = 4 x$ gives $2 x = 6$ , so $x = 3$ .
Define the inverse function of a function.
The inverse function does the opposite of the original function, taking each output back to the input it came from.
Putting a number through a function and then through its inverse gives back the number you started with.
What is the inverse of $input \to \times 6 \to + 4 \to output$ ?
The inverse function is:
$input \to - 4 \to \div 6 \to output$
For example, 5 goes through the original function to give 34, and the inverse takes 34 back to 5.
True or False?
The inverse of the function $\times 2 \to + 3$ is the function $\div 2 \to - 3$ as given here.
False.
The inverse must undo the operations in reverse order, so it is $- 3 \to \div 2$ instead.
For example, the original function takes 1 to 5, and subtracting 3 then halving takes 5 back to 1, whereas halving then subtracting 3 gives $- 0.5$ instead.
Fill in the gap to find the inverse of $x \to \times 4 \to - 5 \to y$ algebraically by making $x$ the subject:
$4 x - 5 = y \Rightarrow x = \frac{______}{4}$
The completed working is:
$4 x - 5 = y \Rightarrow x = \frac{y + 5}{4}$
This shows that the inverse function adds 5 and then divides by 4.
The function $\times 3 \to + 2$ gives an output of 17. Use its inverse to find the input.
The inverse is $- 2 \to \div 3$ , so the input was $(17 - 2) \div 3 = 5$ .
Checking, $5 \times 3 + 2 = 17$ as required.
When you find an inverse function by rearranging its equation to make $x$ the subject, why do you then swap the letters $x$ and $y$ ?
Because after the rearrangement the input is $y$ , and the inverse function is usually wanted with $x$ as its input.
Swapping the letters keeps the same operations but writes them with $x$ going in, so for example $\frac{y - 3}{2}$ becomes $\frac{x - 3}{2}$ .

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