What is a Function? - Algebra I

What is a Function?

In algebra, a function is a special type of relation where each input is related to exactly one output. Functions are fundamental in mathematics and have various applications in real life.

Outline

• Definition of a Function
• Domain and Range
• Function Notation
• Examples and Practice Exercises
• Homework
• Revision
• YouTube Video Explanation

Definition of a Function

A function is a relation where each element of the domain (input) is mapped to exactly one element of the range (output). This can be written as \( f(x) \), where \( f \) is the function and \( x \) is the input.

Domain and Range

The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs.

Function Notation

Functions are often written in the form \( f(x) \), where \( f \) represents the function and \( x \) is the input variable. For example, if \( f(x) = x^2 + 2x + 1 \), then \( f(3) = 16 \).

Examples

Example 1

Given the function \( f(x) = 2x + 3 \), find \( f(4) \).

Solution: \( f(4) = 2(4) + 3 = 11 \).

Example 2

Determine the domain and range of the function \( f(x) = \sqrt{x - 1} \).

Domain: \( x \geq 1 \). Range: \( f(x) \geq 0 \).

Example 3

For the function \( f(x) = x^2 - 4x + 4 \), find \( f(2) \).

Solution: \( f(2) = 2^2 - 4(2) + 4 = 0 \).

Example 4

Find the range of the function \( f(x) = 3x - 5 \) where \( x \) is any real number.

Range: All real numbers, since \( 3x - 5 \) can produce any real number.

Example 5

Evaluate the function \( f(x) = \frac{1}{x - 2} \) at \( x = 3 \).

Solution: \( f(3) = \frac{1}{3 - 2} = 1 \).

Example 6

Determine if \( f(x) = x^2 - 6x + 8 \) is a function and describe its domain and range.

Solution: Yes, it is a function. Domain: All real numbers. Range: \( f(x) \geq 0 \) (parabola opens upwards).

Exercises

Exercise 1

Given the function \( f(x) = x^3 - 2x \), find \( f(-1) \).

Solution: \( f(-1) = (-1)^3 - 2(-1) = -1 + 2 = 1 \).

Exercise 2

Find the domain and range of \( f(x) = \frac{1}{x^2 - 1} \).

Domain: \( x \neq \pm 1 \). Range: All real numbers except 0.

Exercise 3

Evaluate the function \( f(x) = \sqrt{4 - x} \) at \( x = 0 \).

Solution: \( f(0) = \sqrt{4 - 0} = 2 \).

Exercise 4

Find the range of \( f(x) = -2x + 1 \) where \( x \) is any real number.

Range: All real numbers, since \( -2x + 1 \) can produce any real number.

Exercise 5

For the function \( f(x) = x^2 + 2x - 3 \), find \( f(1) \).

Solution: \( f(1) = 1^2 + 2(1) - 3 = 0 \).

Exercise 6

Determine if \( f(x) = \frac{x + 2}{x - 3} \) is a function and describe its domain and range.

Solution: Yes, it is a function. Domain: \( x \neq 3 \). Range: All real numbers except 0.

Homework

Complete the following problems for additional practice:

• Find the value of \( f(x) \) for given inputs in functions similar to the examples provided.
• Determine the domain and range for various functions.
• Explore real-life applications of functions and their notation.

Revision

Review the key concepts of functions:

• A function maps each input to exactly one output.
• The domain is the set of all possible inputs, and the range is the set of all possible outputs.
• Function notation \( f(x) \) represents the value of the function at \( x \).

Video Explanation

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