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Algebraic functions are types of functions that involve variables and constants, using operations such as addition, subtraction, multiplication, division, and exponentiation by a constant.

They can range from simple linear functions like \(f(x) = 2x\) to more complex forms like polynomials and rational functions.

In the given problem, we are dealing with the functions \(f(x) = 2x\) and \(g(x) = 5x - 1\). Both are linear algebraic functions because their highest exponent of x is 1.

To solve this kind of problem, we often use function composition. This involves replacing the variable in one function with another function.

Understanding how to work with algebraic functions and their composition is key to mastering many concepts in algebra and higher-level math.

The substitution method is a technique used to solve problems by replacing one variable with an expression derived from another variable.

For function composition, this means inserting one function into another.
Let's look at the problem: We have \(f(x) = 2x\) and \(g(x) = 5x - 1\). To find \(f(g(x))\), we substitute g(x) into f(x).
So, \(f(g(x)) = f(5x - 1)\). This changes our function from f in terms of x to f in terms of \(5x - 1\).

Next, we use the expression for f(x), which is \(2x\). Thus, \(f(5x - 1) = 2(5x - 1)\).
This method of substitution allows us to combine functions into a single expression, making it easier to work with.

Simplifying expressions is the process of making them more straightforward by performing operations and combining like terms. It makes algebraic problems more manageable and easier to solve.

In our example, after substituting \(5x - 1\) into \(f(x) = 2x\), we get \(f(5x - 1) = 2(5x - 1)\).
To simplify this:

• First, distribute the 2 across the terms inside the parentheses: \[ 2(5x - 1) = 2 \times 5x - 2 \times 1 \]
• This multiplication gives us \[10x - 2 \]

Now we have simplified the composition of the functions to a single expression: \(10x - 2\).

Carrying out these steps helps ensure you correctly find the composition and simplification of algebraic functions.