Functions practice questions (IB Maths AA)

Sample questions

The one free IB Maths AA question on this topic, from the free IB Maths AA SL Paper 1 Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 15 marks · Composite & inverse functions
   The functions $f$ and $g$ are defined by $f(x) = 2x - 3$ for $x \in \mathbb{R}$, and $g(x) = \frac{x + 1}{x - 2}$ for $x \neq 2$.
   $f(x) = 2x - 3, \qquad g(x) = \frac{x + 1}{x - 2}, \quad x \neq 2$
   (a)

   Find $(g \circ f)(3)$.

   [2]
   (b)

   Find $g^{-1}(x)$, stating its domain.

   [3]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →

Key terms

The definitions and formulas IB Maths AA expects for functions, typeset as they appear on the paper.

Composite function

Applying one function after another: $(f \circ g)(x) = f(g(x))$, so $g$ acts first. The domain of $f \circ g$ is the part of the domain of $g$ whose outputs lie in the domain of $f$.

$$(f \circ g)(x) = f\big(g(x)\big)$$

Inverse function

The function that undoes $f$, which exists only when $f$ is one-to-one. Its graph is the reflection of $y = f(x)$ in the line $y = x$, and the domain of $f^{-1}$ is the range of $f$.

$$f\big(f^{-1}(x)\big) = f^{-1}\big(f(x)\big) = x$$

Quadratic function and the discriminant

For $f(x) = ax^2 + bx + c$ the vertex form $a(x - h)^2 + k$ gives the turning point and the axis of symmetry $x = -\frac{b}{2a}$. The discriminant decides the number of real roots: two if positive, one repeated if zero, none if negative.

$$\Delta = b^2 - 4ac, \qquad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Transformations of graphs

Translations, stretches and reflections written as one expression. $d$ translates vertically, $c$ horizontally, $a$ stretches vertically by factor $a$, and $b$ stretches horizontally by factor $\frac{1}{b}$.

$$y = a\,f\big(b(x - c)\big) + d$$

Rational function

The quotient of two linear expressions. Its vertical asymptote is where the denominator is zero and its horizontal asymptote is the ratio of the leading coefficients.

$$f(x) = \frac{ax + b}{cx + d}, \qquad x = -\frac{d}{c}, \qquad y = \frac{a}{c}$$

Exponential and logarithmic functions

$y = a^x$ and $y = \log_a x$ are inverses of each other, with $e^x$ and $\ln x$ the natural pair. $a^x$ has a horizontal asymptote at $y = 0$; $\log_a x$ has a vertical asymptote at $x = 0$.

Frequently asked questions

Other IB Maths AA topics

• Number and algebra
• Geometry and trigonometry
• Statistics and probability
• Calculus

All IB Maths AA topics · IB Maths AA exam guide · All free practice exams