Functions Formulae AI SL & AI HL: Complete IB Mathematics Guide
Functions are the cornerstone of IB Mathematics Applications and Interpretation (AI) at both Standard Level and Higher Level. This comprehensive guide provides all essential formulae, concepts, and practical applications you need to excel in your IB Math AI examinations and coursework. Understanding functions enables you to model real-world phenomena, analyze data patterns, and solve complex mathematical problems with confidence.
Introduction to Functions
A function is a mathematical relationship between two variables where each input value corresponds to exactly one output value. In IB Math AI, functions serve as powerful tools for modeling real-world situations such as population growth, financial investments, temperature changes, and countless other applications. The notation \( f(x) \) represents a function named \( f \) with input variable \( x \), and understanding this notation is fundamental to working with all function types.
Key Function Terminology
Domain: The set of all possible input values (x-values) for which the function is defined. For example, the domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \) because square roots of negative numbers are not real.
Range: The set of all possible output values (y-values) that the function can produce. For the function \( f(x) = x^2 \), the range is \( y \geq 0 \) because squaring any real number always gives a non-negative result.
Composite Functions: When one function is applied to the result of another function, written as \( (f \circ g)(x) = f(g(x)) \). This concept is particularly important for understanding function transformations.
Inverse Functions: A function \( f^{-1}(x) \) that "undoes" the original function \( f(x) \). If \( f(a) = b \), then \( f^{-1}(b) = a \). Not all functions have inverses; only one-to-one functions are invertible.
Linear Functions
Linear functions form the foundation of function study in IB Math AI. These functions create straight-line graphs and model constant rates of change across numerous real-world applications, from calculating taxi fares to predicting business revenue. The simple interest formula is a practical application of linear relationships in finance.
where \( m \) is the gradient (slope) and \( c \) is the y-intercept
where \( a \), \( b \), and \( d \) are constants
where \( (x_1, y_1) \) is a known point and \( m \) is the gradient
calculates the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \)
A taxi charges a base fare of $3.50 plus $2.20 per kilometer. The cost function is \( C(d) = 2.20d + 3.50 \), where \( d \) is distance in kilometers. For a 12 km journey: \( C(12) = 2.20(12) + 3.50 = $29.90 \).
Properties of Linear Functions
- Constant rate of change (gradient remains the same throughout)
- Graph is always a straight line
- Positive gradient means increasing function; negative gradient means decreasing function
- Parallel lines have identical gradients
- Perpendicular lines have gradients whose product is \( -1 \) (i.e., \( m_1 \times m_2 = -1 \))
Quadratic Functions
Quadratic functions produce parabolic curves and appear extensively in physics, engineering, economics, and natural sciences. They model projectile motion, profit optimization, and countless other scenarios where quantities change at variable rates. Understanding quadratic equations is essential for both AI SL and AI HL students.
where \( a \neq 0 \), and \( a \), \( b \), \( c \) are constants
vertical line passing through the vertex of the parabola
where \( (h, k) \) is the vertex (turning point) of the parabola
solves \( ax^2 + bx + c = 0 \) for x-intercepts (roots)
determines the number of real roots: \( \Delta > 0 \) (two roots), \( \Delta = 0 \) (one root), \( \Delta < 0 \) (no real roots)
The discriminant is crucial for analyzing quadratic functions without fully solving them. Always check the discriminant value to determine the nature of solutions before attempting detailed calculations.
| Coefficient Value | Effect on Parabola | Graph Behavior |
|---|---|---|
| \( a > 0 \) | Opens upward | Minimum point at vertex |
| \( a < 0 \) | Opens downward | Maximum point at vertex |
| \( |a| > 1 \) | Vertical stretch | Narrower parabola |
| \( 0 < |a| < 1 \) | Vertical compression | Wider parabola |
Exponential Functions
Exponential functions model growth and decay processes that occur throughout nature, finance, and science. Population growth, radioactive decay, compound interest, and epidemic spread all follow exponential patterns. These functions are characterized by constant percentage rates of change rather than constant absolute changes.
where \( a \) is the initial value and \( b \) is the base (\( b > 0, b \neq 1 \))
where \( e \approx 2.71828 \), \( a \) is initial value, and \( k \) is the growth/decay constant
where FV is future value, PV is present value, \( r \)% is annual rate, \( k \) is compounding periods per year, \( n \) is number of years
Exponential Growth vs Decay
| Type | Condition | Graph Behavior | Real-World Examples |
|---|---|---|---|
| Growth | \( b > 1 \) or \( k > 0 \) | Increases rapidly as x increases | Population growth, compound interest, viral spread |
| Decay | \( 0 < b < 1 \) or \( k < 0 \) | Decreases toward zero as x increases | Radioactive decay, depreciation, cooling |
A bacterial culture doubles every 3 hours. Starting with 500 bacteria, the population after \( t \) hours is \( P(t) = 500 \times 2^{t/3} \). After 9 hours: \( P(9) = 500 \times 2^{9/3} = 500 \times 2^3 = 4000 \) bacteria.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions and play a crucial role in solving exponential equations. They appear in pH calculations, earthquake magnitude (Richter scale), sound intensity (decibels), and numerous scientific applications. Understanding the relationship between exponents and logarithms is fundamental for IB Math AI students.
where \( a > 0 \), \( b > 0 \), and \( a \neq 1 \)
logarithm with base \( e \), the natural base
Laws of Logarithms
useful for calculator computation when base \( a \) is not available
Most calculators only have \( \log_{10} \) (common log) and \( \ln \) (natural log) buttons. Use the change of base formula to calculate logarithms with other bases. For example, \( \log_2(8) = \frac{\ln(8)}{\ln(2)} \).
Function Transformations
Transformations allow you to modify the position, size, and orientation of function graphs without changing their fundamental shape. Understanding transformations is essential for sketching graphs quickly and analyzing how functions behave under various modifications. These concepts apply to all function types studied in IB Math AI.
Types of Transformations
shifts graph \( k \) units vertically (up if \( k > 0 \), down if \( k < 0 \))
shifts graph \( h \) units horizontally (right if \( h > 0 \), left if \( h < 0 \))
stretches if \( |a| > 1 \), compresses if \( 0 < |a| < 1 \)
compresses if \( |b| > 1 \), stretches if \( 0 < |b| < 1 \)
flips graph over the x-axis
flips graph over the y-axis
Rational Functions (AI HL)
Rational functions, which are ratios of polynomial functions, introduce concepts of asymptotes and discontinuities. These functions model relationships with constraints and limitations, such as average cost functions in economics or concentration dilution in chemistry. AI HL students must understand the behavior of rational functions near their asymptotes and discontinuities.
where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \neq 0 \)
vertical asymptote at \( x = h \), horizontal asymptote at \( y = k \)
Asymptotes
- Vertical Asymptotes: Occur at values of \( x \) where \( Q(x) = 0 \) (denominator equals zero). The function approaches \( \pm\infty \) near these values.
- Horizontal Asymptotes: Describe the behavior as \( x \to \pm\infty \). Determined by comparing degrees of numerator and denominator polynomials.
- Oblique Asymptotes: Occur when the numerator's degree is exactly one more than the denominator's degree. Found using polynomial long division.
Piecewise Functions
Piecewise functions are defined by different expressions over different intervals of the domain. These functions model situations where behavior changes at specific thresholds, such as tax brackets, shipping costs based on weight ranges, or cellular phone plans with different rates for different usage levels.
A parking garage charges:
\[ C(t) = \begin{cases} 5 & \text{if } 0 < t \leq 1 \\ 5 + 3(t-1) & \text{if } t > 1 \end{cases} \]where \( t \) is time in hours. First hour costs $5, additional hours cost $3 each.
Composite and Inverse Functions
Composite Functions
apply \( g \) first, then apply \( f \) to the result
Important: Function composition is not commutative. Generally, \( f(g(x)) \neq g(f(x)) \). Always work from the inside out when evaluating composite functions.
Inverse Functions
applying a function and its inverse returns the original value
To find \( f^{-1}(x) \): (1) Replace \( f(x) \) with \( y \), (2) Swap \( x \) and \( y \), (3) Solve for \( y \), (4) Replace \( y \) with \( f^{-1}(x) \). Remember that only one-to-one functions have inverses.
Applications in Real-World Modeling
Functions provide powerful mathematical models for understanding and predicting real-world phenomena. The IB Math AI curriculum emphasizes practical applications across diverse fields, making functions an essential tool for data analysis and decision-making.
Common Applications
- Finance: Compound interest calculations, loan amortization schedules, investment growth projections, and break-even analysis all rely on exponential and linear functions.
- Physics: Projectile motion (quadratic), radioactive decay (exponential), spring oscillations (trigonometric), and wave propagation all use function models.
- Biology: Population dynamics (exponential and logistic functions), enzyme kinetics (rational functions), and growth patterns (various function types).
- Economics: Supply and demand curves (linear and non-linear), cost functions, revenue optimization, and market equilibrium analysis.
- Environmental Science: Temperature changes, pollution concentration, carbon dating, and sustainability modeling.
Interactive Function Calculator
Quadratic Function Calculator
Calculate the vertex and axis of symmetry for quadratic functions in the form \( f(x) = ax^2 + bx + c \)
Study Tips for IB Math AI
Effective Practice Strategies
- Master the Formula Booklet: Familiarize yourself with the official IB formula booklet. Know which formulas are provided and which you need to derive or memorize.
- Practice Graph Sketching: Regular practice sketching function graphs by hand builds intuition about function behavior. Start with basic functions, then add transformations.
- Use Technology Wisely: Learn your GDC (graphing display calculator) thoroughly. Practice verifying analytical solutions with graphical methods.
- Connect Concepts: Understand relationships between different function types. For example, logarithmic functions are inverses of exponential functions.
- Work Through Past Papers: Practice with authentic IB examination questions. Time yourself to build exam stamina and identify areas needing improvement.
- Focus on Application: IB Math AI emphasizes real-world applications. Practice interpreting function models in context and communicating mathematical reasoning clearly.
Common Mistakes to Avoid
- Confusing horizontal and vertical transformations (remember: \( f(x - h) \) moves right, not left)
- Forgetting domain restrictions for logarithmic functions (argument must be positive)
- Incorrectly applying logarithm laws, especially with addition and multiplication
- Mixing up quadratic formula signs when calculating roots
- Neglecting to check whether calculated solutions fall within the given domain
- Forgetting to state units and context when answering application problems
Differences Between AI SL and AI HL
While both levels cover fundamental functions, AI HL extends these concepts with greater depth and additional topics. Understanding these differences helps students set appropriate learning goals and study priorities. For a comprehensive comparison, visit our guide on IB Mathematics AA vs AI.
| Topic Area | AI SL Coverage | AI HL Additional Content |
|---|---|---|
| Linear Functions | Basic forms, gradient, applications | Systems of linear equations, matrix methods |
| Quadratic Functions | Standard and vertex forms, basic applications | Completing the square, optimization problems |
| Exponential Functions | Growth and decay models, compound interest | Continuous compounding, differential equations |
| Logarithmic Functions | Basic laws, simple equations | Change of base, complex logarithmic equations |
| Rational Functions | Basic understanding | Detailed analysis, asymptote determination |
| Function Analysis | Domain, range, basic transformations | Composite functions, inverse functions, advanced transformations |
Additional Resources
Enhance your understanding of functions with these complementary resources available on RevisionTown:
- Number and Algebra Formulae AI SL & AI HL - Master essential algebraic techniques that support function manipulation
- Functions Formulae AA SL & AA HL - Compare with the Analysis and Approaches curriculum
- Arithmetic Sequence Calculator - Practice with sequences and series
- Quadratic Equation Calculator - Verify your quadratic solutions
- Scientific Calculator - Essential tool for function calculations
- Percentage Calculator - Useful for exponential growth/decay calculations
- GPA Calculator - Track your academic progress throughout IB
Exam Preparation Checklist
- Review all function types and their key characteristics
- Practice sketching graphs with and without technology
- Memorize formulas not provided in the formula booklet
- Work through complete past papers under timed conditions
- Verify your GDC is functioning properly and batteries are fresh
- Review common application contexts and how to interpret them
- Practice explaining mathematical reasoning clearly in writing
- Ensure you understand when to use exact vs approximate answers
Conclusion
Mastering functions is fundamental to success in IB Mathematics Applications and Interpretation at both Standard and Higher Levels. This comprehensive guide has covered all essential formulae, from basic linear relationships through complex exponential and logarithmic functions, along with practical applications that demonstrate the power of mathematical modeling. Regular practice with these concepts, combined with effective use of technology and past examination questions, will build the confidence and competence needed to excel in your IB Math AI assessments.
Remember that functions are not isolated mathematical concepts but powerful tools for understanding the world around us. Whether analyzing financial investments, modeling population dynamics, or optimizing business decisions, the function formulae and techniques you have learned provide a foundation for quantitative reasoning that extends far beyond the IB examination room.
Continue exploring additional mathematical topics through our comprehensive collection of IB Mathematics resources, and make use of our various calculators to verify your work and deepen your understanding. Success in IB Math AI comes from consistent practice, deep conceptual understanding, and the ability to apply mathematical knowledge in diverse contexts.
