Further Functions

Summary and Exam Tips for Further Functions

Further Functions is a subtopic of Functions, which falls under the subject Mathematics in the IB DP curriculum. This section explores various types of functions and their properties, focusing on rational functions, solutions of inequalities, modulus functions, and properties of functions such as odd, even, and self-inverse functions.

Rational Functions: These functions can have different numbers of vertical asymptotes depending on their form. For instance, $y = \frac{ax+b}{cx^2+dx+e}$ may have zero, one, or two vertical asymptotes, while $y = \frac{ax^2+bx+c}{dx+e}$ has a vertical asymptote at $x = -\frac{e}{d}$ and an oblique asymptote.

Solutions of Inequalities: Graphical methods are used to solve inequalities involving cubic and modulus functions. By sketching graphs, one can visually determine the solution intervals.

Modulus Functions: To graph $y = |f(x)|$, reflect any negative parts of $y = f(x)$ in the x-axis. For $y = f(|x|)$, reflect the graph of $y = f(x)$ for $x > 0$ in the y-axis.

Properties of Functions: A function is odd if $f(-x) = -f(x)$ and even if $f(-x) = f(x)$. Self-inverse functions satisfy $f^{-1}(x) = f(x)$ and are symmetric with respect to the line $y = x$.

Exam Tips

• Understand Asymptotes: Be clear about how to find vertical, horizontal, and oblique asymptotes for rational functions. Practice sketching these graphs to identify intercepts and asymptotes accurately.

• Graphical Solutions: Use graphical methods to solve inequalities. Sketching graphs can help visualize the solution intervals, especially for complex inequalities.

• Modulus Functions: Practice reflecting graphs for modulus functions. Remember that $y = |f(x)|$ involves reflecting below the x-axis, while $y = f(|x|)$ involves reflecting across the y-axis.

• Function Properties: Familiarize yourself with identifying odd, even, and self-inverse functions. This will help in understanding their symmetry properties and behavior.

• Practice Problems: Regularly solve example problems to reinforce concepts and improve problem-solving speed and accuracy.