Further Functions

Study Notes

Further Functions in mathematics involve understanding the behavior and properties of various types of functions, including rational, modulus, and inverse functions. Key concepts include asymptotes, intercepts, and the graphical representation of inequalities.

• Rational Functions — functions of the form $y = \frac{ax+b}{cx^2+dx+e}$ which may have vertical asymptotes. Example: The function $y = \frac{x-3}{2x^2+x-3}$ has vertical asymptotes at $x = -\frac{3}{2}$ and $x = 1$.
• Modulus Functions — functions involving absolute values, such as $y = |f(x)|$ or $y = f(|x|)$. Example: The graph of $y = |2x-1|$ reflects any negative parts of $y = 2x-1$ in the x-axis.
• Inverse Functions — functions where $f^{-1}(x)$ exists if $f(x)$ is one-to-one. Example: For $f(x) = x^2 + 6x + 4$, the inverse exists for $x \geq -3$.
• Self Inverse Functions — functions where $f^{-1}(x) = f(x)$. Example: The function $f(x) = \frac{x}{x-1}$ is self inverse.

Exam Tips

Key Definitions to Remember

• Rational Function
• Modulus Function
• Inverse Function
• Self Inverse Function

Common Confusions

• Misidentifying vertical asymptotes in rational functions
• Incorrectly reflecting graphs for modulus functions
• Forgetting domain restrictions for inverse functions

Typical Exam Questions

• How do you find the vertical asymptotes of a rational function? Solve $cx^2 + dx + e = 0$ for $x$.
• What is the graphical transformation for $y = |f(x)|$? Reflect parts of $y = f(x)$ below the x-axis.
• How do you determine if a function is self inverse? Check if $f^{-1}(x) = f(x)$.

What Examiners Usually Test

• Ability to sketch graphs of rational and modulus functions
• Understanding of asymptotes and intercepts
• Solving inequalities using graphical methods