Problem 20 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=-4 x ; g(x)=x^{3}+x^{2}-6 $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = -4x^3 - 4x^2 + 24\); \((g \circ f)(x) = -64x^3 + 16x^2 - 6\).

Step 1: Understand Composite Functions

Composite functions involve applying one function to the result of another function. The notation \((f \circ g)(x)\) means that we first apply \(g(x)\), and then apply \(f\) to the result. Similarly, \((g \circ f)(x)\) means we first apply \(f(x)\), and then apply \(g\) to the result.

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

To find \((f \circ g)(x)\), substitute \(g(x) = x^3 + x^2 - 6\) into \(f(x) = -4x\). This gives: \[ (f \circ g)(x) = f(g(x)) = f(x^3 + x^2 - 6). \] Now, replace \(x\) in \(f(x)\) with \(x^3 + x^2 - 6\): \[ f(x^3 + x^2 - 6) = -4(x^3 + x^2 - 6). \] Simplify the expression: \[ (f \circ g)(x) = -4x^3 - 4x^2 + 24. \]

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

To find \((g \circ f)(x)\), substitute \(f(x) = -4x\) into \(g(x) = x^3 + x^2 - 6\). This gives: \[ (g \circ f)(x) = g(f(x)) = g(-4x). \] Now, replace \(x\) in \(g(x)\) with \(-4x\): \[ g(-4x) = (-4x)^3 + (-4x)^2 - 6. \] Simplify each term: \[ (-4x)^3 = -64x^3 \] and \[ (-4x)^2 = 16x^2. \] Thus, \[ (g \circ f)(x) = -64x^3 + 16x^2 - 6. \]

Key Concepts

Function Composition

Function composition is a crucial concept in mathematics that allows you to create a new function by combining two functions. The notation

• \((f \circ g)(x)\) signifies that you apply function \(g\) first and then apply function \(f\) to the result.
• Similarly, \((g \circ f)(x)\) means you apply \(f\) first and then \(g\).

This might sound a little tricky at first, but it's essentially about feeding the output of one function directly into another. This is particularly useful in solving complex mathematical problems where you need to transform a given variable in steps. When you see the circle \(\circ\), remember it stands for "composition" rather than multiplication.
By breaking down complex problems into smaller tasks through composition, mathematics becomes more manageable and efficient.

Algebraic Expressions

An algebraic expression is a blend of numbers, variables, and operators (like + and -). In the given exercise, the function \(g(x) = x^3 + x^2 - 6\) is a perfect example of an algebraic expression.
These expressions can vary from simple to complex and are the building blocks for constructing functions.

• Terms: These are parts of an expression separated by + or - signs.
• Coefficients: Numbers to be multiplied with variables in terms.

Understanding how to manipulate these expressions is key in solving equations and simplifying results, especially when composing functions. By taking one expression and substituting it into another, you can see the power of algebraic manipulation in action. As seen in the solution, substituting \(g(x)\) into \(f(x)\) or the other way around involves careful replacement and basic algebraic operations.

Simplifying Expressions

After you perform function composition, simplifying the resulting expressions is the next vital step.
Simplification involves reducing an expression to its simplest form. The goal is to express the function as clearly as possible with fewer terms or less complicated forms.

• Combine like terms: Group terms with the same power of \(x\).
• Perform arithmetic operations: Calculate exponents, and multiply coefficients.

For example, in the solution for \((f \circ g)(x)\), you simplify \(-4(x^3 + x^2 - 6)\) to \(-4x^3 - 4x^2 + 24\). This process eliminates unnecessary complexity, making the expression easier to work with and understand. Remember, a simplified expression is often more useful for calculating values or deriving further functions in subsequent problems.