OCR GCSE Mathematics (J560) — Algebra
Every time you use a vending machine, pressing a specific button (the input) guarantees a specific snack drops down (the output). A mathematical function works in exactly the same way. It is a mathematical rule or machine that takes a set of inputs (usually represented by ) and transforms them into a set of outputs.
This transformation process is formally known as mapping. The input is the independent variable because you can choose its value, while the output is the dependent variable because its final value depends entirely on what you put into the function. Foundation tier questions often represent this using a physical function machine (a series of boxes containing operations), while Higher tier transitions into formal algebra.
In exams, functions are usually written using function notation, such as . This is read aloud as "f of x". It simply identifies the name of the function (in this case, ) and the specific input variable it requires (). To evaluate a function, you perform , replacing every in the expression with your chosen input value, making sure to follow the correct (BIDMAS).
Given the function , find the output when the input is 7, and find an expression for .
Step 1: To find , substitute the numerical value 7 in place of .
Step 2: Apply the order of operations (multiply before subtracting) to calculate the output.
Step 3: To find , substitute the entire algebraic term in place of .
Step 4: Simplify the expression.
You can tie your shoelaces to secure them, but you must untie them in the exact reverse order to take your shoes off. An inverse function does exactly this mathematically; it "undoes" the operations of the original function. It takes an output and maps it directly back to the original input.
For an inverse to exist, the original rule must be a one-to-one function, meaning every input has exactly one unique output and vice versa. The inverse is written using the notation , which is read as "f inverse of x". On a graph, the curve of an inverse function is always a perfect reflection of the original function across the diagonal line .
To find an inverse function algebraically, you temporarily replace with . You then rearrange the equation to make the subject of the formula. Finally, you swap the and variables to state your final inverse formula in terms of .
Find the inverse function for .
Step 1: Replace with to set up the equation.
Step 2: Multiply both sides by 2 to clear the fraction.
Step 3: Rearrange to isolate the term.
Step 4: Divide by 3 to make the subject of the formula.
Step 5: Swap the and variables to write the final inverse function in terms of .
Why does putting your socks on before your shoes work perfectly, but doing it the other way around creates a mess? The order in which you apply rules matters massively. A composite function is the result of linking two or more functions together, where the output of the first function immediately becomes the input for the next.
The notation means you apply function first, and then apply function to that result. The function closest to the variable is called the inner function and must always be processed first. The function further away is the outer function. OCR sometimes refers to this process as a "succession of two functions".
To find an algebraic expression for , you must substitute the entire algebraic expression for the inner function into every place that appears in the outer function .
Given and , calculate an expression for .
Step 1: Write the outer function , replacing the with empty brackets ready for an input.
Step 2: Substitute the entire inner function into those brackets.
Step 3: Expand the brackets.
Step 4: Simplify the expression to find your final answer.
Using the same functions, and , calculate .
Step 1: Identify the inner function () and evaluate it for the input of 4.
Step 2: Use this result as the input for the outer function ().
Step 3: Calculate the final output.
Students often think means ' multiplied by '. This is completely incorrect; is just the name of the rule, and the in brackets represents the input variable.
When evaluating composite functions like , candidates frequently process them from left to right. You must always apply the inner function (closest to the ) first.
When substituting negative numbers into a function, especially involving powers like , always write the negative number inside brackets (e.g., ) to avoid calculator syntax errors.
For inverse function questions, always ensure your final answer is written in terms of (e.g., ); leaving it in terms of will cost you the final accuracy mark.
In multi-part questions, OCR often awards 'follow-through' marks. If you calculate an incorrect output in part (a), substitute it clearly and correctly into a formula in part (b) to still pick up method marks.
Function
A mathematical rule that assigns exactly one output to each unique input.
Mapping
The formal mathematical process of transforming an input into an output using a specific rule, often written as .
Independent variable
The input value (usually ) that you choose to put into a function.
Dependent variable
The output value (usually or ) whose final result relies on the chosen input.
Function machine
A visual representation of a function using sequential boxes, where each box contains a single mathematical operation.
Function notation
A formal system of writing functions, such as , where is the name of the function and is the input variable.
Substitution
The process of replacing a variable (like ) with a specific numerical value or algebraic expression.
Order of operations
The strict mathematical rules (BIDMAS/BODMAS) dictating the sequence in which calculations must be performed.
Inverse function
A function that performs the opposite operations in the reverse order, mapping an output back to its original input, denoted as .
One-to-one function
A function where every input has a unique output, and every output comes from exactly one unique input.
Subject of the formula
The single variable that stands alone on one side of an equals sign, expressing its value in terms of the other variables.
Composite function
The result of combining two or more functions, where the output of one function becomes the immediate input for the next.
Inner function
In a composite function like , the function closest to the variable which is applied first.
Outer function
In a composite function like , the function applied second, using the result of the inner function as its input.
Put your knowledge into practice — try past paper questions for Mathematics
Function
A mathematical rule that assigns exactly one output to each unique input.
Mapping
The formal mathematical process of transforming an input into an output using a specific rule, often written as .
Independent variable
The input value (usually ) that you choose to put into a function.
Dependent variable
The output value (usually or ) whose final result relies on the chosen input.
Function machine
A visual representation of a function using sequential boxes, where each box contains a single mathematical operation.
Function notation
A formal system of writing functions, such as , where is the name of the function and is the input variable.
Substitution
The process of replacing a variable (like ) with a specific numerical value or algebraic expression.
Order of operations
The strict mathematical rules (BIDMAS/BODMAS) dictating the sequence in which calculations must be performed.
Inverse function
A function that performs the opposite operations in the reverse order, mapping an output back to its original input, denoted as .
One-to-one function
A function where every input has a unique output, and every output comes from exactly one unique input.
Subject of the formula
The single variable that stands alone on one side of an equals sign, expressing its value in terms of the other variables.
Composite function
The result of combining two or more functions, where the output of one function becomes the immediate input for the next.
Inner function
In a composite function like , the function closest to the variable which is applied first.
Outer function
In a composite function like , the function applied second, using the result of the inner function as its input.
