Problem 16 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x-3 ; g(x)=x^{2} $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = x^2 - 3\) and \((g \circ f)(x) = x^2 - 6x + 9\).

Step 1: Understand Function Composition

Function composition involves applying one function to the results of another. For two functions \(f(x)\) and \(g(x)\), there are two compositions: \((f \circ g)(x)\) and \((g \circ f)(x)\).

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

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). Start:\[(f \circ g)(x) = f(g(x)) = f(x^2).\]Since \(f(x) = x - 3\), we have:\[f(x^2) = x^2 - 3.\]Thus, \((f \circ g)(x) = x^2 - 3\).

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

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). Start:\[(g \circ f)(x) = g(f(x)) = g(x-3).\]Since \(g(x) = x^2\), we have:\[g(x-3) = (x-3)^2.\]Apply the square formula:\[(x-3)^2 = x^2 - 6x + 9.\]Thus, \((g \circ f)(x) = x^2 - 6x + 9\).

Key Concepts

Substitution Method

The substitution method is a powerful technique often used in mathematics, particularly when dealing with functions. It involves replacing one variable with another to simplify an expression or solve a problem. In the case of function composition, it helps us find the output of one function when the output of another is inputted into it.

When applying the substitution method:

• Identify the two functions involved: in our example, they are \(f(x) = x - 3\) and \(g(x) = x^2\).
• Choose which function's output will replace the variable in the other function. This tells you the order in which to compose them, either \((f \circ g)(x)\) or \((g \circ f)(x)\).
• Substitute the expression from the second function into the argument of the first function.

This method simplifies the process of understanding how one function influences another by providing clarity when functions are intertwined.

Function Operations

Function operations encompass various methods for combining functions. These operations include addition, subtraction, multiplication, division, and composition. In composition, we use one function's output as the input to another, creating a new function.

To explore function operations with composition, consider:

• Composition: The operation where the output of one function, say \(g(x)\), is used as the input for another, \(f(x)\), resulting in \(f(g(x))\).
• Notation: Composition is denoted by \((f \circ g)(x)\) or \(f(g(x))\), indicating the sequence of operations.
• Order Matters: Remember that \((f \circ g)(x)\) is generally different from \((g \circ f)(x)\). It's important to perform these operations in the specified order.

By understanding these operations, students can effectively navigate situations that require complex function interactions, such as those in algebra and calculus.

Algebraic Manipulation

Algebraic manipulation involves using algebraic techniques to simplify expressions and solve equations. In function composition, manipulation becomes crucial to simplify the composed function and make sense of the resulting expressions.

When working through algebraic manipulation in function composition:

• Simplification: Once functions are composed, simplify the expression by applying algebraic rules. For instance, \((x - 3)^2\) simplifies to \(x^2 - 6x + 9\).
• Distributive Properties: Leverage distributive properties to expand and combine terms effectively. This might involve expanding squares or factoring complex expressions when necessary.
• Combining Like Terms: After manipulation, identify like terms and combine them to achieve the simplest form.

Mastering algebraic manipulation allows you to translate complex compositions into clearer, simpler forms, ultimately making it easier to solve and understand the problem at hand.