Algebra Revision Notes

What is a Function?

A function is a rule that assigns each input (from a set called the domain) to exactly one output (from a set called the range). This 'one-to-one' or 'many-to-one' relationship is crucial. If an input can have more than one output, it is not a function. For example, the relationship from students to their exam scores could define a function (each student gets one score), but the relationship from exam scores to students might not (multiple students could get the same score).

Function Notation: f(x)

Functions are commonly written using f(x) notation. Here, 'f' is the name of the function, and 'x' represents the input variable. So, 'f(x)' means 'the output of the function f when the input is x'. For example, if a function doubles its input and adds three, we can write it as f(x) = 2x + 3. To find the output when the input is 5, we would calculate f(5) = 2(5) + 3 = 10 + 3 = 13.

Evaluating Functions

Evaluating a function means finding its output for a given input. To do this, substitute the input value into the function's expression wherever you see the input variable. For example, given the function g(x) = x² - 4, to find g( -2 ), substitute -2 for x: g( -2 ) = ( -2 )² - 4 = 4 - 4 = 0. Similarly, g(3) = (3)² - 4 = 9 - 4 = 5. Be careful with negative numbers and squaring.

Inverse Operations for Functions

Sometimes you are given the output of a function and need to find the input. This involves solving an equation. For example, if f(x) = 3x - 5 and you are told that f(x) = 10, you need to solve 3x - 5 = 10 for x. Add 5 to both sides: 3x = 15. Then divide by 3: x = 5. So, an input of 5 gives an output of 10 for this function. This is a basic step towards understanding inverse functions later.

What is a Function?

A function, often written as f(x), g(x), or h(x), is a rule that maps each input value (x) to exactly one output value. The 'domain' is the set of all possible input values, and the 'range' is the set of all possible output values. For example, if f(x) = x² + 3, then f(2) = 2² + 3 = 7. You substitute the input value into the expression to find the output. Functions can be represented by equations, graphs, or tables.

Composite Functions

A composite function is formed by applying one function to the result of another function. If you have two functions, f(x) and g(x):

• fg(x) means substitute g(x) into f(x). So, fg(x) = f(g(x)).
• gf(x) means substitute f(x) into g(x). So, gf(x) = g(f(x)).

The order of operations is crucial. Generally, fg(x) will not be the same as gf(x). Always work from the inside out when evaluating composite functions.

Inverse Functions

An inverse function, denoted f⁻¹(x), 'reverses' the action of the original function f(x). If f(a) = b, then f⁻¹(b) = a. Steps to find f⁻¹(x):

1. Write y = f(x).
2. Swap x and y in the equation.
3. Rearrange the new equation to make y the subject.
4. Replace y with f⁻¹(x).

Graphically, the graph of f⁻¹(x) is a reflection of the graph of f(x) across the line y = x. For an inverse function to exist and be a function, the original function must be a one-to-one mapping.

Domain and Range for Inverse Functions

The domain of f(x) becomes the range of f⁻¹(x), and the range of f(x) becomes the domain of f⁻¹(x). When finding an inverse, it's important to consider if the original function is one-to-one. If a function is not one-to-one (e.g., y = x²), you might need to restrict its domain to ensure its inverse is also a function. For example, for f(x) = x² to have an inverse, we might restrict its domain to x ≥ 0, which means its inverse f⁻¹(x) = √x will have a domain of x ≥ 0 and a range of y ≥ 0.

Understanding Sequences and Terms

A sequence is an ordered list of numbers. Each number in the list is called a 'term'. For example, in the sequence 2, 4, 6, 8, ... , the number 2 is the first term, 4 is the second term, and so on. Understanding the terminology is crucial before attempting to generate terms.

Generating Terms using a Term-to-Term Rule

A term-to-term rule tells you how to get from one term to the next. You *must* be given the first term to start. For example, if the first term is 5 and the rule is 'add 4', the sequence terms would be: 5 (first term), 5+4=9 (second term), 9+4=13 (third term), 13+4=17 (fourth term), and so on. You apply the rule repeatedly.

Generating Terms using a Position-to-Term Rule (nth term)

A position-to-term rule (often called an 'nth term rule') expresses each term based on its position in the sequence, usually denoted by 'n'. The first term has a position of n=1, the second term has n=2, the third term has n=3, and so on. To find a term, substitute its position number into the given formula. For example, if the nth term is given by 2n + 1:

• 1st term (n=1): 2(1) + 1 = 3
• 2nd term (n=2): 2(2) + 1 = 5
• 3rd term (n=3): 2(3) + 1 = 7

This allows you to find any term directly without knowing the previous ones.

Key Differences and Common Mistakes

Distinguish between term-to-term and position-to-term rules. A common mistake is to confuse them. If the rule says 'start with 5 and add 3 to get the next term', it's term-to-term. If it says 'the nth term is 3n + 2', it's position-to-term. Always identify the type of rule first. Remember that 'n' always represents the position number (1, 2, 3, ...), not the term value itself.

What is a Sequence?

A sequence is simply an ordered list of numbers, often following a specific pattern or rule. Each number in the sequence is called a 'term'. For example, in the sequence 2, 4, 6, 8, ..., the numbers 2, 4, 6, and 8 are the first, second, third, and fourth terms respectively. Being able to identify the pattern and generate subsequent terms is a core skill.

Generating Terms from a Rule

Often, you'll be given a rule that tells you how to find any term in the sequence. This rule is frequently expressed using 'n', where 'n' represents the position of the term in the sequence. For instance, 'n=1' for the first term, 'n=2' for the second term, 'n=3' for the third term, and so on. To generate terms, you substitute the position number 'n' into the given rule and calculate the result. For example, if the rule is 'nth term = 3n - 1':

• 1st term (n=1): 3(1) - 1 = 3 - 1 = 2
• 2nd term (n=2): 3(2) - 1 = 6 - 1 = 5
• 3rd term (n=3): 3(3) - 1 = 9 - 1 = 8

The sequence generated is 2, 5, 8, ... . Ensure you perform the arithmetic carefully, following order of operations (BIDMAS/PEMDAS).

Types of Sequence Rules

Sequence rules can take various forms. Some common types include:

1. Linear (Arithmetic) Sequences: These have a constant difference between consecutive terms. The rule will typically be in the form 'an + b', where 'a' is the common difference. For example, 5n + 2 generates 7, 12, 17, ... (common difference of 5).
2. Quadratic Sequences: These involve n^2, such as n^2 + 3. The terms are 1^2 + 3 = 4, 2^2 + 3 = 7, 3^2 + 3 = 12, ... .
3. Other Sequences: Rules can also involve indices, fractions, or negative numbers. For example, 2^n generates 2, 4, 8, ... or 1/n generates 1, 1/2, 1/3, ... . Always read the rule carefully and apply it systematically to each term's position.