91Ó°ÊÓ

In algebra, we often work with functions. A function is a relation where each input has a single output.
The function is usually written as \(f(x)\), where \(x\) is the input variable, and the expression defines how to compute the output.
In function substitution, you replace the input variable with another value or expression.
For example, if we have a function \(f(x) = -3x + 4\), and we want to find \(f(-x)\), we substitute \(-x\) for \(x\).
This means everywhere you see \(x\) in the function, you replace it with \(-x\).
Substitution is a fundamental concept because it helps us understand how functions behave when their inputs change
and is widely used in calculus and algebra.

Simplifying expressions means to reduce them to their simplest form. This involves applying arithmetic operations,
removing parentheses, and combining like terms.
In our example, after substituting \(-x\) into the function \(f(x)=-3x+4\), we get \(f(-x)=-3(-x)+4\).
To simplify, we first multiply \(-3\) by \(-x\), which gives us \(+3x\).
The simplified function thus becomes \(+3x + 4\).
Simplification makes functions easier to understand and solve.
Remember always to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

A linear function is a type of function where the graph is a straight line.
The general form of a linear function is \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
In the example function \(f(x) = -3x + 4\), \(-3\) is the slope. The slope determines the steepness and direction of the line.
A positive slope means the line goes upwards, whereas a negative slope means it goes downwards.
The value \(4\) is the y-intercept, which is the point where the line crosses the y-axis.
Linear functions are simple yet powerful tools in algebra, making them widely used for modeling real-world situations.
Understanding linear functions will help you greatly in more advanced math, where functions become more complex.