Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Algebra → Introduction to Functions ↓


Inverse of a Function


In mathematics, especially in algebra, functions play a central role in understanding the relationships between numbers and variables. A function is like a machine that takes input, performs some operation or series of operations on that input, and produces output. But what if we want to reverse the process? This brings us to the concept of the "inverse of a function."

Understanding the tasks

Before diving into the inverse, it is important to understand what a function is. Think of a function as a rule that assigns exactly one output to each input. If we have a function f, and we input a value x, we get an output f(x).

For example, let's consider a simple function:

f(x) = 2x + 3

Here, f takes an input x, multiplies it by 2, and then gets the output by adding 3 to the result.

What is an inverse function?

The inverse function essentially reverses the process of the function. If our original function f takes x to f(x), then the inverse function, usually represented as f-1, will take f(x) back to x.

A function f has an inverse if and only if it is both one-to-one and onto, meaning that each possible output is uniquely paired with exactly one input.

Visual example

Let's visualize this concept. Suppose we have a function that goes from x to y like this:

XYf(x)

The inverse function will look like this:

YXF-1(Y)

Finding the inverse of a function

To find the inverse of a function algebraically, follow these steps:

  1. Replace f(x) with y.
  2. Change the x and y values.
  3. Solve for y.
  4. Replace y with f-1(x).

Let's take the function f(x) = 2x + 3 and find its inverse step by step.

  1. Replace f(x) with y: 
    y = 2x + 3
  2. Replace x and y: 
    x = 2y + 3
  3. Solve for y: 
    x = 2y + 3
     x - 3 = 2y
     y = (x - 3)/2
  4. Replace y with f-1(x): 
    f-1(x) = (x - 3)/2

Therefore, the inverse of the function f(x) = 2x + 3 is f-1(x) = (x - 3)/2.

Verification

To verify our solution, we can check whether f(f-1(x)) = x and f-1(f(x)) = x. This verification confirms that we have found the inverse function correctly.

Example validation:

Check f(f-1(x)):

f(f-1(x)) = f((x - 3)/2)
 = 2((x - 3)/2) + 3
 = x - 3 + 3
 = x

Check f-1(f(x)):

f-1(f(x)) = f-1(2x + 3)
 = ((2x + 3) - 3)/2
 = 2x/2
 = x

Graphical representation

Graphically, a function and its inverse are reflections of each other on the line y = x. Let's illustrate this with the function f(x) = 2x + 3 and its inverse.

Point A (1, 5)Point B (5.5, 15)f(x)f-1(x)

The blue line represents f(x) = 2x + 3, and the red line represents its inverse f-1(x) = (x - 3)/2. The dashed gray line represents the reflection line y = x.

Why the inverse function is important

Inverse functions are essential for solving equations where the value to be determined is the original input before applying the function. They are widely used in real-world applications such as undoing effects, solving equations, and in fields such as computer science, physics, engineering, and economics.

For example, if we have a formula that describes the amount of money in a savings account based on compound interest and we need to find out how long it takes to reach a certain amount, we would use the inverse of the compound interest formula to calculate the time.

Exploring more complex functions

So far, we have worked with linear functions. However, the concept of inverse functions also applies to more complex functions such as quadratic, exponential, and logarithmic functions. Let's explore these:

Quadratic functions:

Consider the quadratic function f(x) = x2. At first glance, finding its inverse seems challenging because the square function is not one-to-one over all real numbers. To handle this, we restrict its domain (e.g., x ≥ 0) to make it one-to-one.

The inverse in this restricted domain is f-1(x) = √x.

Exponential and logarithmic functions:

Exponential functions such as f(x) = ex and their inverses, logarithmic functions such as f-1(x) = ln(x), demonstrate the relationship between growth processes and their reversal mechanisms.

Conclusion

In short, understanding the inverse of a function is important in reversing operations and determining the original inputs when the output is given. From simple linear functions to more complex quadratic and exponential functions, the process of finding the inverse enriches algebraic problem-solving and is a gateway to more advanced mathematical concepts.

By mastering this concept, students gain the tools to tackle a variety of mathematical challenges, connecting concepts from algebra, calculus, and beyond.


Grade 10 → 2.5.4


U
username
0%
completed in Grade 10

← Prev (2.5.3)
Composition of Functions
Next (3) →
Coordinate Geometry

Comments 



Inverse of a Function

https://www.buddymath.com/g10/2.5.4?bmal=en