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Algebraic functions are the cornerstone of algebra and constitute the rules that transform an input into a single output. The classic form is expressed as an equation, like the one you might see in a textbook:

f(x) = 5x+1

In this equation, the function name is 'f', the input variable is 'x', and the equation itself describes how to calculate the output. Think of the function as a special machine: whatever number you feed into it, the machine processes that number according to the rule, which, in this case, multiplies the input by 5 and then adds 1.
The beauty of these functions is their versatility. They can handle simple numbers, fractions, or even other algebraic expressions. To understand these functions, it's crucial to recognize the role of the variable 'x'. It is a placeholder that can be replaced with any value you wish to evaluate.

Function substitution is essentially the process of replacing the variable in a function with a specific value or expression. This is a fundamental skill in algebra that allows us to find out what output a function will give for a given input.
Consider the function provided in the textbook problem:

f(x) = 5x+1

Function substitution involves taking a specific value, such as \(\frac{2}{5}\), and plugging it into the function in place of 'x'. Here's what this looks like in practice:

f\left(\frac{2}{5}\right) = 5\left(\frac{2}{5}\right) + 1

By doing this, we have 'substituted' the value \(\frac{2}{5}\) for 'x'. The next step after making this substitution is to simplify the expression to find out what the function evaluates to for that particular input.

Simplifying expressions is an essential operation in algebra and involves reducing an expression to its most basic form. To simplify an algebraic expression, one must combine like terms, perform arithmetic operations, and reduce fractions if necessary.
Using the textbook example, after substitution we get:

f\left(\frac{2}{5}\right) = 5\left(\frac{2}{5}\right) + 1

To simplify this, we follow the basic arithmetic operation of multiplication first. We multiply 5 by \(\frac{2}{5}\), and since the 5 in the numerator and denominator cancel each other out, it simplifies to 2.

f\left(\frac{2}{5}\right) = 2 + 1

The last step is to add 1 to 2, giving us the simplified result of 3.
By breaking down the expression into these simpler steps, we can easily and systematically find the value of the function for any input, which is a fundamental skill when dealing with algebraic functions.