A function is a special relationship between two sets of numbers or variables. It assigns exactly one output for every input. In simple terms, a function takes an input (often called x), performs a specific operation, and gives an output (often called f(x)).

The function is often written as:

f(x) = expression

Here:

• f(x) is the output of the function, also called the function value.
• x is the input, the value that you plug into the function.
• The expression defines the rule for how to calculate the output.

What Makes a Function?

A function must follow these rules:

• For every x input, there must be exactly one corresponding f(x) output.
• The same input can never be paired with more than one output.

Types of Functions

There are different types of functions based on the relationship between the input and the output. Some common types include:

• Linear Functions: These functions have the form f(x) = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line.
• Quadratic Functions: These functions have the form f(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola.
• Cubic Functions: These functions have the form f(x) = ax³ + bx² + cx + d. The graph of a cubic function has a characteristic "S" shape.

Example 1: Evaluating a Linear Function

Suppose we have the function: f(x) = 2x + 3. To evaluate the function for x = 4:

• Step 1: Substitute x = 4 into the function: f(4) = 2(4) + 3
• Step 2: Simplify: f(4) = 8 + 3 = 11

Therefore, when x = 4, f(x) = 11.

Example 2: Evaluating a Quadratic Function

Suppose we have the quadratic function: f(x) = x² - 4x + 3. To evaluate the function for x = 2:

• Step 1: Substitute x = 2 into the function: f(2) = (2)² - 4(2) + 3
• Step 2: Simplify: f(2) = 4 - 8 + 3 = -1

Therefore, when x = 2, f(x) = -1.

Summary:

A function is a relationship that assigns exactly one output to each input. Functions are commonly written in the form f(x) = expression. The input x is plugged into the expression to find the output f(x). Functions can be linear, quadratic, cubic, or other types, depending on the rule that defines them.

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