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Functions and Algebra Practice (Interactive)

Intuition

Functions are like relationships between variables, they describe how one quantity depends on another: Understanding function behavior, transformations, inverses, and composition, is key to modeling real-world relationships

Why it matters: Functions are the building blocks of mathematical modeling in science, engineering, and economics

The key insight: Understanding function behavior, transformations, inverses, and composition, is key to modeling real-world relationships

IB Maths, Functions and Algebra Practice

8 auto-graded practice problems at medium to hard difficulty, aligned to the IB Mathematics syllabus (SL/HL). Select an answer, submit, and review the explanation.

Worked Examples

Example 1: Function Composition

Problem: If f(x)=2x+1f(x) = 2x + 1 and g(x)=x2−3g(x) = x^2 - 3, find f(g(x))f(g(x)) and g(f(x))g(f(x)).

Solution: Step 1: f(g(x))=f(x2−3)=2(x2−3)+1=2x2−6+1=2x2−5f(g(x)) = f(x^2 - 3) = 2(x^2 - 3) + 1 = 2x^2 - 6 + 1 = 2x^2 - 5

Step 2: g(f(x))=g(2x+1)=(2x+1)2−3=4x2+4x+1−3=4x2+4x−2g(f(x)) = g(2x + 1) = (2x + 1)^2 - 3 = 4x^2 + 4x + 1 - 3 = 4x^2 + 4x - 2

Step 3: Notice f(g(x))≠g(f(x))f(g(x)) \neq g(f(x)) as a rule, composition is not commutative.

Key insight: Always substitute the inner function into the outer function, not the other way around. The order matters.

Example 2: Quadratic Inequality

Problem: Solve x2+2x−8>0x^2 + 2x - 8 > 0.

Solution: Step 1: Factorise: x2+2x−8=(x+4)(x−2)x^2 + 2x - 8 = (x + 4)(x - 2)

Step 2: Find critical points: x=−4x = -4 and x=2x = 2

Step 3: Test intervals:

  • x<−4x < -4: (−)(−)=(+)(-)(-) = (+) ✓
  • −4<x<2-4 < x < 2: (+)(−)=(−)(+)(-) = (-) ✗
  • x>2x > 2: (+)(+)=(+)(+)(+) = (+) ✓

Step 4: Answer: x<−4x < -4 or x>2x > 2

Key insight: For >> or << inequalities, use strict inequalities. For ≥\geq or ≤\leq, include the critical points.

Example 3: Geometric Series

Problem: Find the sum of the first 8 terms of a geometric series with first term 5 and common ratio 2.

Solution: Step 1: Formula: Sn=arn−1r−1S_n = a \frac{r^n - 1}{r - 1} (for r≠1r \neq 1)

Step 2: Substitute: S8=5×28−12−1=5×256−11=5×255=1275S_8 = 5 \times \frac{2^8 - 1}{2 - 1} = 5 \times \frac{256 - 1}{1} = 5 \times 255 = 1275

Step 3: Verify: terms are 5, 10, 20, 40, 80, 160, 320, 640. Sum = 1275 ✓

Key insight: The geometric sum formula Sn=arn−1r−1S_n = a\frac{r^n - 1}{r - 1} works for any r≠1r \neq 1. For ∣r∣<1|r| < 1, the sum to infinity is S∞=a1−rS_\infty = \frac{a}{1 - r}.

Functions

Algebra

Sequences

Binomial Theorem

Common Mistakes

Confusing the binomial coefficient with the term itself: C(n,k) is just the coefficient. The full term includes the powers of a and b: C(n,k)a^(n-k)b^k. Don’t forget the powers.

Forgetting to check domain restrictions: When solving equations or finding inverses, always check for values that make expressions undefined (division by zero, square roots of negatives).

Mixing up function composition with multiplication: f(g(x)) is composition, not f(x) × g(x). The order matters: f(g(x)) ≠ g(f(x)) as a rule.

Cross-References