Unit 4 Class 10 Math New Book Solutions
Unit 4 Class 10 Math New Book Solutions are available here in easy, step-by-step PDF format. Unit 4 is titled Functions and Graphs and includes Exercise 4.1, Exercise 4.2, Exercise 4.3, and Review Exercise 4.
This unit introduces functions, operations on functions, composite functions, inverse functions, absolute value graphs, absolute value equations, and inequalities. It also explains how these mathematical ideas are used in practical situations.
The solutions include complete working, graphs, number lines, examples, and checked final answers. Students can use these notes for classwork, homework, tests, revision, and exam preparation.
Table of Contents
- Exercise-wise Solutions of Unit 4
- Quick Overview of Unit 4
- What Is Unit 4 About?
- Solutions of Exercise 4.1
- Solutions of Exercise 4.2
- Solutions of Exercise 4.3
- Solutions of Review Exercise 4
- Important Formulas of Unit 4
- Common Mistakes in Unit 4
- Related Class 10 Math Resources
- FAQs About Unit 4 Solutions
- Disclaimer
- Final Words
Exercise-wise Solutions of Unit 4 Class 10 Math
Open an exercise below to view its complete step-by-step PDF solutions. The exercises are arranged in the same order as the Class 10 Mathematics new book.
Exercise-wise Solutions of Unit 4 Class 10 Math
Quick Overview of Unit 4 Class 10 Math New Book
Exercise 4.1
Operations on functions, composite functions, inverse functions, domain, range, and function evaluation.
Exercise 4.2
Absolute value graphs, equations, inequalities, intervals, and number-line representation.
Exercise 4.3
Real-life applications of functions and absolute value inequalities.
Review Exercise 4
MCQs, function operations, inverse functions, graphs, inequalities, and practical applications.
What Is Unit 4 Class 10 Math New Book About?
Unit 4 is titled Functions and Graphs. It explains how one quantity can depend on another quantity. This relationship is represented mathematically by a function.
A function receives an input and produces a related output. The input is usually represented by x, while the result is written as f(x).
To find the value of the function when x = 4, substitute 4 in place of x.
f(4) = 2(4) + 3 = 11Students learn how to add, subtract, multiply, and divide two functions. They also study composite functions, where the output of one function becomes the input of another function.
Another major topic is the inverse of a function. An inverse function reverses the action performed by the original function.
The second part of the unit explains absolute value functions. Their basic graphs are V-shaped. Students learn how to find the vertex, plot points, solve absolute value equations and inequalities, and represent solutions on number lines.
The final exercise connects functions and absolute value inequalities with practical situations involving money, travelling time, temperature limits, measurements, and manufacturing tolerances.
Solutions of Exercise 4.1 Class 10 Math New Book
Exercise 4.1 introduces the main algebraic concepts of functions. Students perform operations on functions, calculate composite functions, find inverse functions, and study domain and range.
- Addition of functions
- Subtraction of functions
- Multiplication of functions
- Division of functions
- Composite functions
- Evaluation of functions
- Inverse functions
- Verification of inverse functions
- Domain and range
- Equality of two functions
Operations on Functions
If f(x) and g(x) are two functions, their sum is:
(f + g)(x) = f(x) + g(x)Their difference is:
(f − g)(x) = f(x) − g(x)Their product is:
(fg)(x) = f(x)g(x)Their quotient is:
(f/g)(x) = f(x)/g(x)In the quotient of two functions, values that make g(x) = 0 must be excluded.
Composite Functions
A composite function is formed when one function is placed inside another function.
(f ∘ g)(x) = f(g(x))This means that g is applied first and f is applied afterwards.
(g ∘ f)(x) = g(f(x))The order is important because f(g(x)) and g(f(x)) may produce different results.
Inverse Functions
The inverse of f(x) is written as f⁻¹(x). To find an inverse function:
- Write y = f(x).
- Rearrange the equation to make x the subject.
- Interchange x and y.
- Write the result as f⁻¹(x).
Domain and Range of an Inverse
The domain and range exchange their positions when an inverse function is formed.
Domain of f⁻¹ = Range of f Range of f⁻¹ = Domain of fThis rule can be used even when the complete inverse function has not been calculated.
Verification of an Inverse Function
A function and its inverse should cancel each other. Therefore:
f(f⁻¹(x)) = x f⁻¹(f(x)) = xIf both expressions simplify to x, the inverse function has been verified.
Solutions of Exercise 4.2 Class 10 Math New Book
Exercise 4.2 focuses on absolute value functions, their graphs, equations, inequalities, interval notation, and number-line representation.
- Plotting absolute value graphs
- Finding the vertex
- Horizontal and vertical shifts
- Absolute value equations
- Absolute value inequalities
- Interval notation
- Open and closed endpoints
- Number-line representation
Graphs of Absolute Value Functions
The basic absolute value function is:
y = |x|Its graph is V-shaped and has its vertex at:
(0, 0)A transformed absolute value function may be written as:
y = a|x − h| + kThe vertex of this graph is:
(h, k)Effect of the Value of a
The value of a affects the direction and steepness of the graph.
- If a is positive, the graph opens upward.
- If a is negative, the graph opens downward.
- A larger value of |a| produces a narrower graph.
- A smaller positive value of |a| produces a wider graph.
In |x + 3|, the graph shifts three units to the left. Therefore, the x-coordinate of the vertex is −3.
Absolute Value Equations
For an equation of the form |A| = k, where k is positive, two separate cases are formed:
A = k or A = −kAbsolute Value Inequalities
Values Between Two Limits
|A| ≤ k ⇒ −k ≤ A ≤ k |A| < k ⇒ −k < A < kThese forms normally give one interval between two endpoints.
Values Outside Two Limits
|A| ≥ k ⇒ A ≥ k or A ≤ −k |A| > k ⇒ A > k or A < −kThese forms normally produce two separate intervals.
Number-Line Representation
- Use a closed circle when an endpoint is included, as with ≤ or ≥.
- Use an open circle when an endpoint is not included, as with < or >.
- Shade between two points when all values inside an interval are included.
- Draw outward rays when the solution contains values outside two endpoints.
Solutions of Exercise 4.3 Class 10 Math New Book
Exercise 4.3 explains how functions and absolute value inequalities are used in practical situations. Students convert information from word problems into mathematical expressions and substitute the required values.
Applications of Functions
- Calculating bank balances
- Finding distance-based fares
- Calculating product costs
- Finding travelling time
- Calculating printing charges
Applications of Absolute Value
- Finding safe temperature ranges
- Finding acceptable product dimensions
- Checking machine-part alignment
- Applying manufacturing tolerances
- Setting quality-control limits
Using Functions in Practical Situations
A practical situation can often be represented by a function. The constant part normally represents a fixed amount, while the variable part changes according to time, distance, quantity, or another measurement.
Here, Rs. 5000 is the starting balance and Rs. 200 is added every month. For six months:
B(6) = 5000 + 200(6) B(6) = 6200Therefore, the total balance after six months is Rs. 6200.
Students should identify what each variable represents, substitute the given value carefully, simplify the expression, and write the final answer with its correct unit.
Absolute Value in Measurement and Safety
Absolute value inequalities are useful when a quantity must remain within, or outside, a fixed distance from a central value.
This means that the temperature differs from 37°C by more than 2.5°C.
T < 34.5°C or T > 39.5°CTherefore, the process must be stopped below 34.5°C or above 39.5°C.
The acceptable rod length is from 2.46 m to 2.54 m.
Solutions of Review Exercise 4 Class 10 Math New Book
Review Exercise 4 combines the major concepts studied throughout the unit. It begins with multiple-choice questions and continues with written questions involving functions, graphs, inverse functions, equations, inequalities, and practical applications.
Conceptual and Algebraic Questions
- Function notation
- Evaluation of functions
- Operations on functions
- Composite functions
- Inverse functions
- Domain restrictions
Graphs and Applications
- Absolute value graphs
- Vertical line test
- Absolute value equations
- Absolute value inequalities
- Profit and discount functions
- GPS acceptable ranges
The review exercise is useful for final revision because it tests both mathematical understanding and the ability to apply formulas in practical situations.
Students should attempt Review Exercise 4 after completing Exercises 4.1, 4.2, and 4.3.
Important Formulas of Unit 4
Operations on Functions
(f + g)(x) = f(x) + g(x) (f − g)(x) = f(x) − g(x) (fg)(x) = f(x)g(x) (f/g)(x) = f(x)/g(x), where g(x) ≠ 0Composite Functions
(f ∘ g)(x) = f(g(x)) (g ∘ f)(x) = g(f(x))Inverse Functions
f(f⁻¹(x)) = x f⁻¹(f(x)) = x Domain of f⁻¹ = Range of f Range of f⁻¹ = Domain of fAbsolute Value Graph
y = a|x − h| + k Vertex = (h, k)Absolute Value Equation
|A| = k ⇒ A = k or A = −kAbsolute Value Inequalities
|A| ≤ k ⇒ −k ≤ A ≤ k |A| < k ⇒ −k < A < k |A| ≥ k ⇒ A ≥ k or A ≤ −k |A| > k ⇒ A > k or A < −kCommon Mistakes in Unit 4
Related Class 10 Math Resources
FAQs About Unit 4 Class 10 Math New Book Solutions
What is the name of Unit 4 in the Class 10 Math new book?
The name of Unit 4 is Functions and Graphs.
How many exercises are included in Unit 4?
Unit 4 contains Exercise 4.1, Exercise 4.2, Exercise 4.3, and Review Exercise 4.
What topics are covered in Exercise 4.1?
Exercise 4.1 covers operations on functions, composite functions, inverse functions, function evaluation, domain, and range.
What topics are covered in Exercise 4.2?
Exercise 4.2 covers absolute value graphs, equations, inequalities, interval notation, and number lines.
What is studied in Exercise 4.3?
Exercise 4.3 contains practical applications involving fares, costs, travelling time, temperature, measurements, and manufacturing.
What is the difference between f⁻¹(x) and 1/f(x)?
f⁻¹(x) represents the inverse function, while 1/f(x) represents the reciprocal of a function. They are different mathematical concepts.
What shape does an absolute value graph have?
The graph of the basic absolute value function y = |x| is V-shaped.
Are graphs included in the Unit 4 solutions?
Yes. The PDF solutions include absolute value graphs and number-line representations wherever required.
Disclaimer
These Unit 4 Class 10 Math New Book Solutions are prepared for educational, learning, and revision purposes. Every effort has been made to provide correct, clear, and easy solutions.
Students should also consult their official mathematics textbook and follow the instructions of their teachers. Notes of Math is not affiliated with the textbook publisher or any educational board.
Final Words
Unit 4 is important because it connects algebra, graphs, inequalities, and practical applications. Students learn how functions work, how functions can be combined, and how inverse functions reverse an operation.
The unit also develops graphing and inequality-solving skills through absolute value functions. Exercise 4.3 shows how these mathematical ideas are used in finance, travelling, measurement, manufacturing, temperature control, and quality assurance.
Students should practise function notation, composite functions, inverse functions, graph vertices, interval notation, and number-line representation carefully. The complete PDFs provided above can be used for step-by-step learning and exam revision.