Functions and graphs

Summary and Exam Tips for Functions and graphs

Functions and graphs is a subtopic of Pure Mathematics 3, which falls under the subject Mathematics in the Edexcel International A Levels curriculum. This chapter covers essential concepts such as the modulus function, functions and mapping, composite functions, inverse functions, and transformations like $y = |f(x)|$ and $y = f(|x|)$. The modulus function is defined as $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$, and is used to solve equations like $|ax + b| = k$. Functions and mapping involve understanding one-one and many-one functions, determining ranges, and verifying graphically. Composite functions involve combining two functions, while inverse functions reverse the operation of a function, existing only for one-one functions. Transformations such as $y = |f(x)|$ involve reflecting parts of graphs, and combining transformations requires maintaining the order of operations. Finally, solving modulus problems involves algebraic and graphical methods to find solutions to inequalities like $|x + 2| < |3x + 1|$.

Exam Tips

• Understand Key Terms: Make sure you are familiar with terms like domain, range, one-one function, and inverse function. These are foundational for solving problems in this chapter.
• Graphical Verification: Always verify your answers graphically where possible. This is particularly useful for checking the range of functions and the correctness of inverse functions.
• Composite Functions: Practice combining functions and ensure you understand how to find the domain and range of composite functions.
• Modulus Problems: Be comfortable with both algebraic and graphical methods to solve modulus equations and inequalities.
• Transformations: Pay attention to the order of transformations when combining them, as this can affect the final result.