Вивчайте Algebraic Functions | Functions and Their Properties

An algebraic function is any function that can be expressed using basic arithmetic operations and variables. These functions form the backbone of mathematical modeling, allowing you to make predictions and understand patterns.

Types of Algebraic Functions

There are 6 types of functions used in data science, namely:

1. Identity Functions

Form: $f(x) = x$ ;
Use case: representing unchanged data or as a reference in transformations.

2. Constant Functions

Form: $f(x) = k$ ;
Use case: representing fixed quantities like flat fees or baseline values.

3. Linear Functions

Form: $f(x) = mx + b$ ;
Use case: predicting outcomes such as revenue or costs.

4. Polynomial Functions

Form: $f(x) = a_nx^n + a_{n - 1}x^{n - 1} + ... + a_1x + a_0$ ;
Use case: regression models, curve fitting, and describing real-world phenomena like parabolic motion.

5. Rational Functions

Form:

f(x) = \frac{p(x)}{q(x)};

Use case: modeling constrained systems, such as rates of change or resource usage.

Behavior of Functions in Python

Now that you've seen these functions in action, here's a summary of their behavior:

1. Identity Function

Equation: $f(x) = x$

Behavior:

Passes through the origin $(0,0)$ ;
A straight line with a slope: $m = 1$ ;
Every input maps to itself;
No maximum or minimum;
Domain: $(-\infty, \infty)$ ;
Range: $(-\infty, \infty)$ .

2. Constant Function

Equation: $f(x) = c$

Behavior:

A horizontal line at: $y = c$ ;
The function output remains constant regardless of input;
Slope: $m = 0$ ;
No maximum or minimum;
Domain: $(-\infty, \infty)$ ;
Range: $\{c\}$ .

3. Linear Function

Equation: $f(x) = mx + b$

Behavior:

A straight line with slope: $m$ ;
If $m > 0$ the function is increasing; if $m < 0$ the function is decreasing.
X-intercept:

x = -\frac{b}{m};

Y-intercept: $y = b$ ;
No maximum or minimum;
Domain: $(-\infty, \infty)$ ;
Range: $(-\infty, \infty)$ .

4. Polynomial Function (Quadratic Example)

Equation: $f(x) = ax^2 + bx + c$

Behavior:

Parabolic curve (U-shaped if $a > 0$ ; inverted U if $a < 0$ );
Has a vertex at:

x = -\frac{b}{2a};

X-intercepts (roots) found using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a};

Y-intercept: $f(0) = c$ ;
Domain: $(-\infty, \infty)$ ;
Range:
- If $a > 0$ then $[y_{\ vertex}; \infty)$ ;
- If $a < 0$ then $(-\infty; y_{\ vertex}]$ .

5. Rational Function

Equation:

f(x) = \frac{1}{x - 1};

Behavior:

Vertical asymptote at: $x-1$ ;
Horizontal asymptote at: $y = 0$ ;
Undefined at $x = 1$ ;
Rapidly decreases and increases near the vertical asymptote;
Domain: $(-\infty, 1) \cup (1, \infty), (-\infty, 0) \cup (0, \infty)$ .

1. Write a Python function that represents the following polynomial function and evaluate it at $x = 4$ :

f(x) = 5x^3 - 2x^2 + 7x - 3

2. What is the output of $f(x) = 2x + 3$ when $x = 4$ ?

Write a Python function that represents the following polynomial function and evaluate it at $x = 4$ :

f(x) = 5x^3 - 2x^2 + 7x - 3

Select the correct answer

def polynomial_function(x):
    return 5 * x**3 - 2 * x**2 + 7 - 3
print(polynomial_function(4))

def polynomial_function(x):
    return 5 * x**3 - 2 * x + 7 * x - 3
print(polynomial_function(4))

def polynomial_function(x):
    return 5 * x**3 - 2 * x**2 + 7 * x - 3
print(polynomial_function(4))

def polynomial_function(x):
    return 5 * x**2 - 2 * x**2 + 7 * x - 3
print(polynomial_function(4))

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Types of Algebraic Functions

There are 6 types of functions used in data science, namely:

1. Identity Functions

Form: $f(x) = x$ ;
Use case: representing unchanged data or as a reference in transformations.

2. Constant Functions

Form: $f(x) = k$ ;
Use case: representing fixed quantities like flat fees or baseline values.

3. Linear Functions

Form: $f(x) = mx + b$ ;
Use case: predicting outcomes such as revenue or costs.

4. Polynomial Functions

Form: $f(x) = a_nx^n + a_{n - 1}x^{n - 1} + ... + a_1x + a_0$ ;
Use case: regression models, curve fitting, and describing real-world phenomena like parabolic motion.

5. Rational Functions

Form:

f(x) = \frac{p(x)}{q(x)};

Use case: modeling constrained systems, such as rates of change or resource usage.

Behavior of Functions in Python

Now that you've seen these functions in action, here's a summary of their behavior:

1. Identity Function

Equation: $f(x) = x$

Behavior:

Passes through the origin $(0,0)$ ;
A straight line with a slope: $m = 1$ ;
Every input maps to itself;
No maximum or minimum;
Domain: $(-\infty, \infty)$ ;
Range: $(-\infty, \infty)$ .

2. Constant Function

Equation: $f(x) = c$

Behavior:

A horizontal line at: $y = c$ ;
The function output remains constant regardless of input;
Slope: $m = 0$ ;
No maximum or minimum;
Domain: $(-\infty, \infty)$ ;
Range: $\{c\}$ .

3. Linear Function

Equation: $f(x) = mx + b$

Behavior:

A straight line with slope: $m$ ;
If $m > 0$ the function is increasing; if $m < 0$ the function is decreasing.
X-intercept:

x = -\frac{b}{m};

Y-intercept: $y = b$ ;
No maximum or minimum;
Domain: $(-\infty, \infty)$ ;
Range: $(-\infty, \infty)$ .

4. Polynomial Function (Quadratic Example)

Equation: $f(x) = ax^2 + bx + c$

Behavior:

Parabolic curve (U-shaped if $a > 0$ ; inverted U if $a < 0$ );
Has a vertex at:

x = -\frac{b}{2a};

X-intercepts (roots) found using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a};

Y-intercept: $f(0) = c$ ;
Domain: $(-\infty, \infty)$ ;
Range:
- If $a > 0$ then $[y_{\ vertex}; \infty)$ ;
- If $a < 0$ then $(-\infty; y_{\ vertex}]$ .

5. Rational Function

Equation:

f(x) = \frac{1}{x - 1};

Behavior:

Vertical asymptote at: $x-1$ ;
Horizontal asymptote at: $y = 0$ ;
Undefined at $x = 1$ ;
Rapidly decreases and increases near the vertical asymptote;
Domain: $(-\infty, 1) \cup (1, \infty), (-\infty, 0) \cup (0, \infty)$ .

1. Write a Python function that represents the following polynomial function and evaluate it at $x = 4$ :

f(x) = 5x^3 - 2x^2 + 7x - 3

2. What is the output of $f(x) = 2x + 3$ when $x = 4$ ?

Write a Python function that represents the following polynomial function and evaluate it at $x = 4$ :

f(x) = 5x^3 - 2x^2 + 7x - 3

Select the correct answer

def polynomial_function(x):
    return 5 * x**3 - 2 * x**2 + 7 - 3
print(polynomial_function(4))

def polynomial_function(x):
    return 5 * x**3 - 2 * x + 7 * x - 3
print(polynomial_function(4))

def polynomial_function(x):
    return 5 * x**3 - 2 * x**2 + 7 * x - 3
print(polynomial_function(4))

def polynomial_function(x):
    return 5 * x**2 - 2 * x**2 + 7 * x - 3
print(polynomial_function(4))

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Секція 1. Розділ 4

Algebraic Functions

Types of Algebraic Functions

1. Identity Functions

2. Constant Functions

3. Linear Functions

4. Polynomial Functions

5. Rational Functions

Behavior of Functions in Python

1. Identity Function

2. Constant Function

3. Linear Function

4. Polynomial Function (Quadratic Example)

5. Rational Function

1. Write a Python function that represents the following polynomial function and evaluate it at x=4x = 4x=4:

2. What is the output of f(x)=2x+3f(x) = 2x + 3f(x)=2x+3 when x=4x = 4x=4?

Awesome!

Algebraic Functions

Types of Algebraic Functions

1. Identity Functions

2. Constant Functions

3. Linear Functions

4. Polynomial Functions

5. Rational Functions

Behavior of Functions in Python

1. Identity Function

2. Constant Function

3. Linear Function

4. Polynomial Function (Quadratic Example)

5. Rational Function

1. Write a Python function that represents the following polynomial function and evaluate it at x=4x = 4x=4:

2. What is the output of f(x)=2x+3f(x) = 2x + 3f(x)=2x+3 when x=4x = 4x=4?

1. Write a Python function that represents the following polynomial function and evaluate it at $x = 4$ :

2. What is the output of $f(x) = 2x + 3$ when $x = 4$ ?

1. Write a Python function that represents the following polynomial function and evaluate it at $x = 4$ :

2. What is the output of $f(x) = 2x + 3$ when $x = 4$ ?