Problem 88 - Algebra 2

Question

If \((f \circ g)(x)=x^{2}-6 x+8\) and \(g(x)=x-3,\) which expression could represent \(f(x) ?\) $$\begin{array}{llll}{\text { F. } x-4} & {\text { G. } x-1} & {\text { H. } x^{2}-1} & {\text { J. } x^{2}-6 x+5}\end{array}$$

Step-by-Step Solution

Answer

Unfortunately, \(f(x) = x^2 -12x + 27\) is not an option in the multiple choice answers. There is an error in the question or the provided options.

Step 1: Substitute g(x) in the function

Substitute \(g(x) = x-3\) into \((f \circ g)(x)=x^{2}-6 x+8\). This will turn \((f \circ g)(x)\) into \(f(x-3)\).

Step 2: Find f(x)

This gives us \(f(x-3)= (x-3)^{2}-6 (x-3)+8\). By solving the equation, it simplifies to \(f(x-3) = x^{2}-6 x+9-6 x+18+8 = x^2 - 12x + 27\).

Step 3: Replace the variable to get the function f(x)

Replace the variable \(x-3\) in \(f(x-3)=x^{2}-12x+27\) with \(x\) to get the function \(f(x) = x^2 -12x + 27\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that involve the combination of variables and constants using a finite number of arithmetic operations: addition, subtraction, multiplication, division, and involution (raising to a power). In the context of the exercise, we are dealing with a composition of functions, which involves using one function as an input for another. This concept allows us to analyze complex relationships by breaking them down into simpler ones.

• The function composition \((f \circ g)(x)\) means applying function \(g\) first and then applying function \(f\) to the result.
• Understanding how to decompose and manipulate these functions is key to solving problems involving algebraic functions.

When approaching algebraic function problems, such as the one given here, it is crucial to pay attention to the order in which functions are applied. This involves substituting the given function into another precisely and dealing with the algebra involved systematically.

Substitution Method

The substitution method involves replacing a variable with a specific expression in an equation. In this exercise, substitution is a critical step to finding the function \(f(x)\). The process follows a logical sequence:

• Substitute \(g(x) = x-3\) into the composite function \((f \circ g)(x) = x^{2}-6x+8\). This substitution transforms the equation into \(f(x-3)\).
• The next move is resolving the expression \(f(x-3) = (x-3)^2 - 6(x-3) + 8\). Simplifying this results in the equation \(f(x-3) = x^2 - 12x + 27\).
• The crucial final step is switching the expression \(x-3\) back to \(x\), thus identifying the function \(f(x) = x^2 - 12x + 27\).

Substitution not only aids in simplifying complex expressions but also assists in transforming equations for further analyses. Mastery of this method provides a powerful tool for solving various algebraic functions.

Polynomial Functions

Polynomial functions, among the most straightforward types of algebraic functions, consist of multiple terms, each having variables raised to whole-number powers with coefficients. They appear frequently in mathematics due to their versatility and the breadth of problems they can model.

• The polynomial form seen in the problem is the quadratic equation \(x^2 - 12x + 27\), which is a representation of \(f(x)\).
• Quadratic polynomials have a maximum degree of 2, with their general formula given as \(ax^2 + bx + c\).
• These functions have significant properties like parabolas in geometry, which are applicable in various fields including physics and engineering.

Understanding the structure, behavior, and properties of polynomial functions is necessary for solving complex polynomial equations. Learning to identify and manipulate these functions forms the backbone of more advanced studies in calculus and algebra.