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 $f(x) = 2x + 1$ and $g(x) = x^2 - 3$, find $f(g(x))$ and $g(f(x))$.

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

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

Step 3: Notice $f(g(x)) \neq g(f(x))$ 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 $x^2 + 2x - 8 > 0$.

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

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

Step 3: Test intervals:

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

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

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: $S_n = a \frac{r^n - 1}{r - 1}$ (for $r \neq 1$)

Step 2: Substitute: $S_8 = 5 \times \frac{2^8 - 1}{2 - 1} = 5 \times \frac{256 - 1}{1} = 5 \times 255 = 1275$

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

Key insight: The geometric sum formula $S_n = a\frac{r^n - 1}{r - 1}$ works for any $r \neq 1$. For $|r| < 1$, the sum to infinity is $S_\infty = \frac{a}{1 - r}$.

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

• Number and Algebra: Algebra supports all practice
• Functions: Functions are tested
• Calculus: Calculus practice covers core techniques

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.