Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find \((\mathbf{a})\) \(\boldsymbol{f} \circ \boldsymbol{g},(\mathbf{b}) \boldsymbol{g} \circ \boldsymbol{f},\) and, if possible, \((\mathbf{c})(\boldsymbol{f} \circ \boldsymbol{g})(\mathbf{0}).\) $$f(x)=2 x^{2}, \quad g(x)=x+4$$

Step-by-Step Solution

Answer

The solutions are \(f \circ g = 2x^2 + 16x + 32\), \(g \circ f = 2x^2 + 4\), and \((f \circ g)(0) = 32\).

Step 1: Compute \(f \circ g\)

Following the definition of function composition, \(f \circ g\) means we substitute \(g\) into \(f\). Hence, \(f(g(x)) = 2 \cdot (x + 4)^2\).

Step 2: Simplify \(f \circ g\)

To simplify \((x + 4)^2\), we'll expand it: \((x + 4)^2 = (x^2 + 8x + 16)\). Substituting this into \(f(g(x))\), we obtain \(f(g(x)) = 2 \cdot (x^2 + 8x + 16)\). Further simplification gives \(f(g(x)) = 2x^2 + 16x + 32\).

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

Next, we'll find \(g \circ f\). This means we substitute \(f\) into \(g\). So, \(g(f(x)) = 2x^2 + 4\).

Step 4: Find \((f \circ g)(0)\)

Finally, substitute \(0\) into \(f \circ g\). Hence, \((f \circ g)(0) = 2(0)^2 + 16(0) + 32 = 32\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables, each multiplied by a coefficient. In the exercise, the function \( f(x) = 2x^2 \) is a polynomial function. It is a quadratic polynomial because the highest power of \( x \) is 2.
Polynomials are crucial in algebra due to their broad range of applications, from simple calculations to modeling complex real-life phenomena. They are easy to manipulate and can represent various types of relationships.

• **Coefficients**: The numbers that multiply the powers, like 2 in \( 2x^2 \).
• **Degree**: The highest power of \( x \), which is 2 in this example.

Polynomial functions like these are the building blocks for understanding more advanced topics in mathematics. Understanding how to work with them is essential for solving real-world problems using algebra.

Function Operations

Function operations include various methods to combine functions. One key operation is **composition of functions**, which is the primary focus of the exercise.
Composition involves plugging one function into another, denoted by \( f \circ g \) or \( g \circ f \) in this exercise. Here:

• \( f \circ g(x) = f(g(x)) \): Plug \( g(x) = x + 4 \) into \( f(x) = 2x^2 \).
• \( g \circ f(x) = g(f(x)) \): Plug \( f(x) = 2x^2 \) into \( g(x) = x + 4 \).

The result is new functions that combine the operations of \( f \) and \( g \), providing insights into their collective behavior. Such operations are fundamental in mathematics because they allow us to explore complex relationships by using simpler, well-understood functions.

Mathematical Problem-Solving

Mathematical problem-solving is a methodical process of finding solutions to problems, often involving multiple steps, as demonstrated in the exercise. The problem asks for the composition of two functions and evaluates them at a given point, zero.
The key steps include:

• **Understanding the Problem**: Identify the functions involved and what operations are required.
• **Decomposition and Recomposition**: Perform the composition \( f \circ g \) and \( g \circ f \) by substituting.
• **Simplification**: Expand and simplify the resulting functions as shown in the solution steps.
• **Evaluation**: Calculate specific values like \((f \circ g)(0)\) by substituting \( x = 0 \) into the simplified function.

Problem-solving in mathematics relies on logical reasoning, analysis, and sometimes creativity. Breaking problems into smaller tasks helps manage complex problems, making them more approachable and understandable.