Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Functions and Graphs → Inverse Functions ↓


Restrictions for Inverses


When we study functions and their inverses, one of the important concepts we learn is the idea of restrictions for an inverse function. The inverse function essentially reverses the operation performed by the original function. If we have a function f(x), then its inverse, denoted as f -1 (x), will satisfy this condition:

f(f - 1 (x)) = x

And

f - 1 (f(x)) = x

However, not every function has an inverse. To have an inverse, the function must be bijective, which means it must be both one-to-one (injective) and onto (surjective). If a function is not bijective, we often apply restrictions to make it so. Let's dive deeper into this topic by discussing these requirements and how restrictions help us define the inverse.

Understanding one-to-one functions

A function is called one-to-one (or injective) if each element of the function's domain corresponds to a unique element in the codomain. This means that two different inputs do not result in the same output. Mathematically, the function f(x) is called one-to-one if:

If f(x 1 ) = f(x 2 ), then x 1 = x 2

For example, consider the following function:

Visual example:

x 1x 2Y

In this illustration, each point on the domain (x-axis) maps to a different point on the codomain (y-axis), ensuring that the function is one-to-one.

Understanding the onto function

If every possible output value in the co-domain is mapped by at least one input value from the domain, then the function is said to be onto (or surjective). This means that the range of the function is equal to its co-domain. Mathematically, the function f(x) is onto if for every element y in the co-domain, there exists an x in the domain such that:

f(x) = y

Why do functions need to be binary?

For a function to have an inverse, it must be both one-to-one and onto: a bijection. This is necessary because each y value must correspond to a unique x value and vice versa. If a function is not bijection, some y values may correspond to multiple x values, making it impossible to define a unique inverse, while some y values may have no x values mapped to them, thus lacking an inverse image.

Introduction to restrictions using domain and range

If a function is not naturally bijective, we can impose restrictions on its domain or codomain to make it bijective. This often involves restricting the domain of the function so that it becomes one-to-one and onto over the restricted part.

Example 1: Square function

Consider the function f(x) = x 2 The graph of this function is a parabola, which is not one-to-one because each positive y value corresponds to two different x values (one positive and one negative). For example:

f(x) = 4, where x can be 2 or -2

Consider this function graphically:

Visual example:

f(x) = x 2

To make this function one-to-one, we can restrict its domain to the non-negative numbers [0, ∞). Then, each y value has a unique x value, and we can find the inverse, which is the square root function:

f -1 (y) = √y, y ≥ 0

Example 2: Sine function

Consider the sine function f(x) = sin(x), which is periodic and neither one-to-one nor finite over its entire domain. To find its inverse, the arcsine function, we restrict the domain to [-π/2, π/2], where the sine function is one-to-one and finite over its range [-1, 1].

Consider this function graphically:

Visual example:

-π/2π/2f(x) = sin(x)

The restricted function is now suitable for defining the inverse:

f -1 (x) = arcsin(x), -1 ≤ x ≤ 1

Common challenges and solutions

Working with inverse functions often brings a number of challenges. Let's take a look at some common problems and their solutions:

Issue: Non-bijective functions

Solution: Restrict the domain to the region where the function is bijective. Use knowledge of the function's behavior to formulate suitable restrictions that allow for a reasonable inverse.

Issue: Identifying the correct domain for the inverse

Solution: Analyze the function graphically and mathematically to understand where it is one-to-one. Consider symmetry and periodicity for trigonometric functions and apply transformations to further simplify expressions.

Issue: Complex inverse expressions

Solution: Simplify the inverse function by taking advantage of algebraic identities and properties. Break down complex expressions into basic components to create easier solutions.

Other examples

Example 3: Exponential and logarithmic functions

Exponential functions and their inverses provide another useful example. Consider f(x) = a x, where a > 1. This function is one-to-one over its entire domain of R and its range of (0, ∞), making it bijective without the need for restrictions.

The inverse is the logarithm of the base a, which is expressed as:

f -1 (x) = log a (x), for x > 0

Example 4: Quadratic function

Quadratic functions often require domain restrictions for their inverses. As shown earlier, the original quadratic function f(x) = x 2 must be restricted to ensure a one-to-one mapping. The inverse of a restricted quadratic can then be expressed in terms of radicals.

Example 5: Cubic function

Consider a cubic function f(x) = x 3 This function is naturally one-to-one and onto over its domain and range. Unlike quadratic functions, cubic functions generally do not require restrictions for the inverse.

The inverse function is straightforward:

f -1 (x) = x 1/3

Conclusion

In short, the process of finding the inverse of a function requires a full understanding of its properties and behavior. While the inverses of some functions can be obtained easily, others require restrictions to conform to the conditions of bijection.

By analyzing domain and range characteristics, applying thoughtful restrictions, and taking advantage of algebraic identities, students can successfully determine inverses, thereby increasing their knowledge and problem-solving abilities in mathematics.

Continuing to explore inverses for different functions and in different contexts helps to further master this essential topic. Remember, practice in identifying and applying restrictions is crucial in the journey to mastering inverse functions.


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Restrictions for Inverses

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