Problem 9 - Applied Calculus

Question

Find the following: (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=2 x+3\( and \)g(x)=5 x^{2}$$

Step-by-Step Solution

Answer

(a) \( f(g(x)) = 10x^2 + 3 \); (b) \( g(f(x)) = 20x^2 + 60x + 45 \); (c) \( f(f(x)) = 4x + 9 \).

Step 1: Understand the Functions

We are given two functions: \( f(x) = 2x + 3 \) and \( g(x) = 5x^2 \). The goal is to find compositions of these functions in different orders.

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

To find \( f(g(x)) \), substitute the expression for \( g(x) \) into \( f(x) \). Since \( g(x) = 5x^2 \), we have:\[ f(g(x)) = f(5x^2) = 2(5x^2) + 3 \]Simplify the expression:\[ f(g(x)) = 10x^2 + 3 \]

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

To find \( g(f(x)) \), substitute the expression for \( f(x) \) into \( g(x) \). Since \( f(x) = 2x + 3 \), we have:\[ g(f(x)) = g(2x + 3) = 5(2x + 3)^2 \]Now expand the square:\[ (2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2 = 4x^2 + 12x + 9 \]Substitute back:\[ g(f(x)) = 5(4x^2 + 12x + 9) \]Distribute 5:\[ g(f(x)) = 20x^2 + 60x + 45 \]

Step 4: Calculate \( f(f(x)) \)

To find \( f(f(x)) \), substitute \( f(x) \) into itself:\[ f(f(x)) = f(2x + 3) = 2(2x + 3) + 3 \]Simplify inside the parenthesis:\[ f(f(x)) = 4x + 6 + 3 \]Further simplify:\[ f(f(x)) = 4x + 9 \]

Key Concepts

Composite Functions

Composite functions are an integral concept in mathematics, particularly when dealing with function operations. A composite function involves combining two functions to create a new function. It is essentially a function applied within another function.
For instance, if you have two functions, \( f(x) \) and \( g(x) \), the notation \( f(g(x)) \) represents the composite function where \( g(x) \) is plugged into \( f(x) \). To achieve this, you take the output of \( g(x) \) and use it as the input for \( f(x) \). It's like following a sequence of operations to get a final result.
This concept can be confusing at first, but it follows a straightforward logical process. You start by finding \( g(x) \), substitute it into \( f(x) \), and solve step-by-step. Understanding composite functions opens doors to solving complex mathematical problems by breaking them into smaller, manageable parts.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to find desired solutions. It is a skill required for successfully solving composite functions.
When working with composite functions, algebraic manipulation helps you simplify the expressions. For example, replacing the variable in \( f(x) \) with \( g(x) \). This requires understanding how basic operations and properties work:

• Substitution: Replace variables with given expressions.
• Distributive Property: Multiply across parentheses correctly.
• Combining Like Terms: Simplify expressions by grouping similar terms.

Algebraic manipulation not only provides clarity on how the composite functions operate but also enables more straightforward solving of expressions by reducing them to simpler forms. When you substitute and manipulate correctly, solving compositions becomes more intuitive.

Polynomial Functions

Polynomial functions are a type of mathematical function that are represented by polynomials. These functions are comprised of variables raised to whole number exponents and include terms with coefficients.
Considering our original problem, both functions \( f(x) = 2x + 3 \) and \( g(x) = 5x^2 \) are polynomial functions. The former is linear since it has an exponent of 1, while the latter is quadratic due to its exponent of 2. When composing these, it's important to understand how to operate them as polynomials.
Each polynomial maintains its characteristics through various operations such as addition, multiplication, or composition. For example, the composition \( f(g(x)) = 10x^2 + 3 \) results in another polynomial which combines elements from both original functions.
Polynomial functions are crucial as they provide a clear framework within which function composition operates, showcasing how different equations interact and yield new, often more complex polynomials.