Many equations can be difficult to solve explicitly, but much easier to understand if we ask a different question: how many solutions are possible?

In this chapter, you will learn to interpret solutions as intersections of graphs and identify the critical values at which the number of intersections can change. You will begin with familiar turning points, then extend the same reasoning to modulus functions, moving lines, periodic functions and restricted domains.

The key is to look for the structural events that create, remove or merge solutions. These may be turning points, vertices, tangent or parallel positions, endpoints, extreme values, or changes in a repeating pattern.

Rather than solving an equation separately for every possible parameter value, you will learn to find the edge cases first, divide the parameter into meaningful regions, and count the solutions in each region.

The main goal is to develop a general method for counting solutions from the structure of a graph, even when familiar algebraic methods such as the quadratic discriminant are no longer enough.

Understanding Function Relationships

In many function questions, you are given a formula such as $f(x)=x^2+3$ and can calculate a value directly.

But some questions give you a relationship instead. For example, $f(2x+1)=x^2+3$.

If you want to find $f(9)$, you first find when $2x+1=9$.

This gives $x=4$, so $f(9)=4^2+3=19$.

You do not need to find a formula for $f(x)$ first.

Other questions may involve two functions, such as $f(g(x))$, or tell you that a function is one-to-one. In these questions, you need to keep track of which value is the input to each function and whether you can work backwards from an output to an input.

You will also see cases where two different inputs give the same output. For example, $f(x)=(x-3)^2$ gives the same value at $x=2$ and $x=4$. So from $f(a)=f(b)$, you cannot always conclude that $a=b$.

In this chapter, you will practise:

• finding a required function value from a given relationship;
• working with composite functions such as $f(g(x))$;
• working backwards when a function is one-to-one;
• recognising when more than one input can give the same output;
• using restrictions on the input to choose between possible values;
• applying functions defined by rules given in the question.

The aim is to decide exactly what the given relationship tells you before doing any unnecessary algebra.

Functional Equations and Recurrence Relations

In this chapter, you will work with functions and sequences that are defined by relationships rather than by a single formula.

For example, instead of being given a formula for $f(x)$, you might be told that $f(x)+2f(3-x)=x+6$.

You can use the relationship with different inputs to find particular values of the function or to prove properties of the function. You usually do not need to find a complete formula for $f$.

You will also meet relationships that can be applied repeatedly. An input may return to an earlier value after several steps, or a sequence may begin to repeat. Recognising this can turn a question involving many steps into a much shorter calculation.

For recurrence relations, each term is defined using one or more earlier terms. You will learn when to calculate a few terms to find a pattern, when to look for a repeating cycle, when to rewrite the recurrence in a simpler form, and when it is easier to work backwards from a later term.

By the end of this chapter, you should be able to:

• choose useful inputs in a functional equation;
• use one relationship more than once to find unknown function values;
• recognise fixed points and repeating cycles;
• use a cycle to find a term with a large index;
• recognise when a recurrence becomes simpler after rewriting it;
• work backwards through a recurrence when a later term is given.

The main skill is knowing which value to substitute or calculate next, rather than trying to find a general formula when the question does not require one.