# 7 Lecture

## Operations on Functions

In calculus and analytical geometry, operations on functions are used extensively to solve problems and understand the behavior of functions.

## Important Mcq's Midterm & Finalterm Prepration Past papers included

1. What is the composition of two functions f and g? A. f(x) + g(x) B. f(x)g(x) C. f(g(x)) D. g(f(x)) Solution: C

2. What is the domain of the function f(x) = 1/x? A. all real numbers except 0 B. all real numbers C. all positive real numbers D. all negative real numbers Solution: A

3. Which of the following is an example of a polynomial function? A. f(x) = 1/x B. f(x) = x^2 + 3x - 5 C. f(x) = ?x D. f(x) = e^x Solution: B

4. What is the range of the function f(x) = sin(x)? A. [-1, 1] B. (-?, ?) C. [0, 1] D. [-?/2, ?/2] Solution: A

5. What is the inverse of the function f(x) = 2x - 3? A. f^-1(x) = x/2 + 3/2 B. f^-1(x) = 2x + 3 C. f^-1(x) = (x - 3)/2 D. f^-1(x) = 3 - x/2 Solution: C

6. Which of the following is an example of an odd function? A. f(x) = x^2 B. f(x) = x^3 C. f(x) = sin(x) D. f(x) = cos(x) Solution: B

7. What is the difference between the domain and range of a function? A. There is no difference. B. The domain is the set of all input values, while the range is the set of all output values. C. The domain is the set of all output values, while the range is the set of all input values. D. The domain and range are the same things. Solution: B

8. What is the equation of the line that passes through points (1, 2) and (3, 4)? A. y = 2x - 1 B. y = x + 1 C. y = 2x + 1 D. y = x - 1 Solution: D

9. What is the composite function of f(x) = x^2 and g(x) = x + 1? A. f(g(x)) = (x + 1)^2 B. f(g(x)) = x^2 + 1 C. g(f(x)) = x^2 + 1 D. g(f(x)) = (x + 1)^2 Solution: A

10. What is the degree of the polynomial function f(x) = 3x^4 + 2x^3 - 5x^2 + 7? A. 0 B. 2 C. 3 D. 4 Solution: D

## Subjective Short Notes Midterm & Finalterm Prepration Past papers included

1. What is the domain of a function? Answer: The domain of a function is the set of all input values (or independent variables) for which the function is defined.

2. What is the range of a function? Answer: The range of a function is the set of all output values (or dependent variables) that the function can produce.

3. What is the difference between a composite function and a simple function? Answer: A simple function is a function that consists of a single equation, while a composite function is a function that is formed by combining two or more functions.

4. What is the inverse of a function? Answer: The inverse of a function is a new function that reverses the operation of the original function.

5. What is the difference between a one-to-one function and a many-to-one function? Answer: A one-to-one function is a function that maps each element of the domain to a unique element of the range, while a many-to-one function is a function that maps multiple elements of the domain to a single element of the range.

6. What is the composition of functions? Answer: The composition of functions is the process of combining two or more functions to create a new function.

7. What is the difference between a domain and a codomain? Answer: The domain of a function is the set of all input values, while the codomain is the set of all possible output values.

8. What is a linear function? Answer: A linear function is a function that can be represented by a straight line on a graph.

9. What is a polynomial function? Answer: A polynomial function is a function that can be represented by a polynomial equation, which is an equation that involves only addition, subtraction, and multiplication of variables raised to whole number powers.

10. What is the difference between an even function and an odd function? Answer: An even function is a function that is symmetric about the y-axis, meaning that f(x) = f(-x) for all values of x. An odd function is a function that is symmetric about the origin, meaning that f(x) = -f(-x) for all values of x.

### Operations on Functions

Calculus is a branch of mathematics that deals with the study of continuous change. One of the fundamental concepts in calculus is the notion of a function. A function is a mathematical object that associates each element of a set (called the domain) with a unique element of another set (called the range). In calculus and analytical geometry, operations on functions are used extensively to solve problems and understand the behavior of functions. One of the basic operations on functions is addition. Given two functions f(x) and g(x), their sum is defined as the function h(x) = f(x) + g(x). The domain of h is the intersection of the domains of f and g. The range of h is the set of all values of f(x) + g(x) for x in the domain of h. The addition of functions is commutative and associative. That is, f(x) + g(x) = g(x) + f(x) and (f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x)). Another important operation on functions is multiplication. Given two functions f(x) and g(x), their product is defined as the function h(x) = f(x) * g(x). The domain of h is the intersection of the domains of f and g. The range of h is the set of all values of f(x) * g(x) for x in the domain of h. Multiplication of functions is commutative and associative. That is, f(x) * g(x) = g(x) * f(x) and (f(x) * g(x)) * h(x) = f(x) * (g(x) * h(x)). The inverse of a function is another important operation in calculus and analytical geometry. Given a function f(x), its inverse, denoted by f^-1(x), is a function that “undoes” the operation of f. That is, if y = f(x), then x = f^-1(y). The inverse of a function exists only if the function is one-to-one. A function is one-to-one if it maps distinct elements of its domain to distinct elements of its range. The graph of the inverse of a function is the reflection of the graph of the function across the line y = x. Composition of functions is another important operation in calculus and analytical geometry. Given two functions f(x) and g(x), their composition is defined as the function h(x) = f(g(x)). The domain of h is the set of all x such that g(x) is in the domain of f. The range of h is the set of all values of f(g(x)) for x in the domain of h. Composition of functions is not commutative. That is, f(g(x)) is not the same as g(f(x)) in general. The derivative of a function is another important operation in calculus. The derivative of a function f(x) at a point x is defined as the limit of the difference quotient as the second point approaches the first point. That is, f'(x) = lim (h->0) (f(x+h) - f(x))/h. The derivative of a function measures how fast the function is changing at a given point. The derivative of a function can be used to find the slope of the tangent line to the graph of the function at a given point. Integration is another important operation in calculus. Integration is the reverse operation of differentiation. Given a function f(x), its indefinite integral is a family of functions F(x) that have f(x) as their derivative. That is, F'(x) = f(x) for all x in the domain of F(x). The definite integral of a function f(x) from a