Problem 15 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x^{2}+1 ; g(x)=5 x $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 25x^2 + 1\) and \((g \circ f)(x) = 5x^2 + 5\).

Step 1: Understand Composition of Functions

When we are asked to find \((f \circ g)(x)\), it means we need to substitute \(g(x)\) into \(f(x)\). Similarly, \((g \circ f)(x)\) requires \(f(x)\) to be substituted into \(g(x)\). Let's start with \((f \circ g)(x)\).

Step 2: Compute \((f \circ g)(x)\)

To find \((f \circ g)(x)\), we substitute \(g(x) = 5x\) into \(f(x)\). This means replace every \(x\) in \(f(x) = x^2 + 1\) with \(5x\): \[f(g(x)) = (5x)^2 + 1 = 25x^2 + 1.\]

Step 3: Compute \((g \circ f)(x)\)

Now, to find \((g \circ f)(x)\), we substitute \(f(x) = x^2 + 1\) into \(g(x)\). This means replace every \(x\) in \(g(x) = 5x\) with \(x^2 + 1\):\[g(f(x)) = 5(x^2 + 1) = 5x^2 + 5.\]

Key Concepts

Function Substitution

In mathematics, function substitution is the process of inputting one function into another. This is a core part of understanding the composition of functions.
When you see notation like \((f \circ g)(x)\), it means you're substituting the output of function \(g(x)\) into the function \(f(x)\). Similarly, \((g \circ f)(x)\) means placing the output of \(f(x)\) into \(g(x)\). This can be thought of as a two-step process where one function acts as the input into another.

• For \((f \circ g)(x)\): Substitute \(g(x) = 5x\) into \(f(x) = x^2 + 1\).
• For \((g \circ f)(x)\): Substitute \(f(x) = x^2 + 1\) into \(g(x) = 5x\).

This simple idea of substitution allows us to manipulate functions in interesting ways and find new function combinations.

Function Operation

Function operation refers to the different ways functions can be combined or manipulated to produce new functions. In the given exercise, we're mainly working with the operation known as composition of functions.

This operation is like plugging the output of one function directly into the input of another. It differs from typical operations like addition or multiplication in that it creates a whole new function by transforming inputs through multiple functions.

• For \((f \circ g)(x)\), you first evaluate \(g(x)\) and then use its result within \(f(x)\).
• For \((g \circ f)(x)\), you first evaluate \(f(x)\) and use the result as the input for \(g(x)\).

This ability to sequence functions is powerful in modeling complex systems in mathematics and can often simplify otherwise complicated expressions.

Algebraic Manipulation

Algebraic manipulation involves transforming expressions to simplify them or find their value. In the context of function composition, these manipulations allow us to derive the final forms of composed functions neatly.

After substituting one function into another, algebraic simplification becomes necessary to express the composed function in a standard form.

• For \(f(g(x)) = (5x)^2 + 1\), we square \(5x\) to get \(25x^2\), then add 1, resulting in \(25x^2 + 1\).
• For \(g(f(x)) = 5(x^2 + 1)\), distribute the 5 across \(x^2 + 1\) using the distributive property: \(5x^2 + 5\).

Simplifying expressions like these is a fundamental skill in algebra, important for ensuring that the functions are clear, concise, and ready for further analysis or application.