FUNCTIONS

Click for photo source

A mathematical function is a relation between a set of inputs and a set of possible outputs with the property that each input is associated with exactly one output. In other words, a function is a rule or a formula that assigns a unique output to each input.

Functions are often represented using mathematical notation, such as f(x), where f is the name of the function and x is the input variable. The output of the function is denoted by f(x), which is read as "f of x".

Functions can take various forms, including algebraic functions, trigonometric functions, logarithmic functions, and many others. They are widely used in various fields of mathematics, as well as in science, engineering, economics, and other areas, to model real-world phenomena, analyze data, and solve problems.

Uses of Mathematical Functions:

Mathematical functions have numerous practical applications in various fields, including:

Science and Engineering: Mathematical functions are used to model physical phenomena, such as motion, heat transfer, fluid dynamics, and electrical circuits. They are also used to solve differential equations, which are fundamental in modeling many physical processes.
Economics and Finance: Mathematical functions are used in financial modeling, such as calculating interest rates, investment returns, and risk management. They are also used in economic modeling, such as forecasting demand and supply, pricing strategies, and production optimization.
Computer Science and Information Technology: Mathematical functions are used in computer algorithms, such as sorting and searching, encryption, and compression. They are also used in data analysis, such as regression analysis, clustering, and pattern recognition.
Statistics and Probability: Mathematical functions are used to calculate statistical measures, such as mean, variance, and correlation. They are also used to model probability distributions, such as the normal distribution, the binomial distribution, and the Poisson distribution.
Mathematics Education: Mathematical functions are fundamental concepts in mathematics education, helping students to develop critical thinking skills, problem-solving skills, and logical reasoning abilities.

Mathematical functions are used to model, analyze, and solve problems in various fields, making them essential tools for understanding and manipulating the natural and man-made world around us.

Solved Problems on Mathematical Functions:

1. Find the value of f(x) = 2x + 5 when x = 3.

Solution:

$f(x) = 2x + 5 ,\ when \ {\color{Red} x = 3}$

To find the value of f(x) when x = 3, we simply substitute x = 3 into the expression for f(x):

$f(x) = 2x + 5$

$f({\color{Red} 3}) = 2({\color{Red} 3}) + 5$

$f({\color{Red} 3}) = 6 + 5$

$f({\color{Red} 3}) = 11$ , answer

2. Find the domain of the function below.

$f(x) = \sqrt{9 - x^2}$

Solution:

The function f(x) is defined as the square root of the expression (9 - x^2), which means that the value under the square root must be non-negative, or:

$9 - x^2 >= 0$

$x^2 \leq 9$

Taking the square root of both sides, we get:

$|x| \leq 3$ , answer

3. Find the inverse function of f(x) = 2x - 3.

Solution:

To find the inverse function of f(x), we need to switch the roles of x and f(x) and solve for x:

$Let \ y = f(x) = 2x - 3$

Switching the roles of x and f(x), we get:

$x = 2y - 3$

$x {\color{Red} +3}= 2y - 3{\color{Red} +3}$

$x {\color{Red} +3}= 2y$ , or

$2y =x {\color{Red} +3}$

$({\color{Red} 2}y =x +3{})*\frac{1}{{\color{Red} 2}}$

$y=\frac{x+3}{2}$

Therefore, the inverse function of f(x) is,

$g(x) = \frac{x + 3 }{2}$ , answer

4. Find the roots of the quadratic function below.

$f(x) = x^2 - 4x + 3$

Solution:

To find the roots of the function f(x), we need to solve the equation f(x) = 0.

$x^2 - 4x + 3 = 0$

Factorizing the quadratic expression, we get:

$(x - 3)(x - 1) = 0$

Therefore,

x =3 and 1 , answer

5. Find the maximum value of the function below

$f(x) = -x^2 + 4x + 1$

Solution:

To find the maximum value of the function f(x), we need to find the vertex of the parabola. The x-coordinate of the vertex is given by x = -b/2a, where a and b are the coefficients of the quadratic expression f(x) = ax^2 + bx + c. In this case, a = -1 and b = 4, so we have:

$x = \frac{-b}{2a} = \frac{-4}{2*(-1)} = 2$

Substituting x = 2 into the expression for f(x), we get:

$f(2) = -2^2 + 4(2) + 1 = 5$

Therefore, the maximum value of the function f(x) is 5, which occurs at x = 2.

@ x = 2

$f(x) = -x^2 + 4x + 1=-2^2 + 4(2) + 1=5$ , answer

6. Find the derivative of the function below.

$f(x) = x^3 - 2x^2 + x - 1$

Solution:

To find the derivative of the function f(x), we need to differentiate each term of the expression separately using the power rule of differentiation.Click to learn how.

$f(x) = x^3 - 2x^2 + x - 1$

Taking the first derivative we get,

$f'(x) = 3x^2 - 4x + 1$

Therefore, the derivative of the function f(x) is

$f'(x) = 3x^2 - 4x + 1.$ , answer

7. Find the limit of the function below, as x approaches 2.

$f(x) = \frac{x^2 - 4 }{x - 2}$

Solution:

To find the limit of the function f(x) as x approaches 2, we can simply substitute x = 2 into the expression for f(x) and evaluate the result:

$f(x) = \frac{x^2 - 4 }{x - 2}$

As x approaches to 2,

$f(2) = \frac{2^2 - 4 }{2-2} = \frac{4-4}{0}=undefined$

However, we can see that the denominator of the expression (x - 2) approaches zero as x approaches 2. Therefore, we can factorize the numerator of the expression and simplify as follows:

$f(x) = \frac{x^2 - 4 }{x - 2} = \frac{({\color{Red} x-2})(x+2)}{{\color{Red} x-2}}=x+2$

Therefore,

So x approaches to 2,

$f(2) =x+2=2+2=4$ , answer

8. Solve the equation f(x) = g(x), where f(x) = 2x + 1 and g(x) = 3x - 5.

Solution:

To solve the equation f(x) = g(x), we need to equate the expressions for f(x) and g(x) and solve for x:

$f(x) = g(x)$

Substituting each equivalent we get,

$2x + 1 = 3x - 5$

Simplifying the equation, we get:

$2x{\color{Red} -2x} + 1 {\color{Blue} +5}= 3x{\color{Red} -2x} - 5{\color{Blue} +5}$

$x=6$ , answer

Therefore, the solution to the equation f(x) = g(x) is x = 6.

9. Find the inverse function of f(x) = log(base 2) (x + 1).

Solution:

To find the inverse function of f(x), we need to switch the roles of x and f(x) and solve for x:

$Let \ y = f(x) = log(base 2) (x + 1)$

Switching the roles of x and f(x), we get:

$x = 2^y - 1$

Solving for y, we get:

$y = log(base 2) (x + 1)$ , answer

10. Find the average rate of change of the function f(x) = $3x^2-2x-1$ on the interval [1, 2].

Solution:

To find the average rate of change of the function f(x) on the interval [1, 2], we need to calculate the difference in the values of the function at the endpoints of the interval and divide by the difference in the values of x at the endpoints:

$Average \ rate \ of \ change = \frac{(f(2) - f(1)) }{(2 - 1)}$