Problem 10 - Algebra 2

Question

Let \(f(x)=3 x+5\) and \(g(x)=x^{2} .\) Perform each function operation. $$ (f \cdot g)(x) $$

Step-by-Step Solution

Answer

\((f \cdot g)(x) = (3x + 5) \cdot x^{2}\)

Step 1: Understand the Multiplication of Functions

The multiplication of functions \( f \) and \( g \) is defined as: \((f \cdot g)(x) = f(x) \cdot g(x)\). So, in this exercise, you need to compute the function \( f(x) \) and multiply it by \( g(x) \).

Step 2: Compute \(f(x)\) and \(g(x)\)

We are given the functions as \( f(x) = 3x + 5 \) and \( g(x) = x^{2} \). We don't need to do anything further with these functions in their current form.

Step 3: Compute \((f \cdot g)(x)\)

To get \((f \cdot g)(x)\), substitute \(f(x)\) and \(g(x)\) into the formula for the multiplication of functions. This results in: \((f \cdot g)(x) = (3x + 5) \cdot x^{2}\).

Key Concepts

Multiplication of Functions

When you multiply two functions, say \( f(x) \) and \( g(x) \), you use the multiplication of functions formula: \((f \cdot g)(x) = f(x) \cdot g(x)\). This can be interpreted as finding the product of the values of each function at a specific point \( x \). In the context of our exercise:- You start with \( f(x) = 3x + 5 \) and \( g(x) = x^2 \).- To find the product \( (f \cdot g)(x) \), you simply take \( f(x) \) and multiply it by \( g(x) \). For this exercise, the calculation would be:- \( (f \cdot g)(x) = (3x + 5) \cdot x^2 \).- Distribute \( x^2 \) to each term inside the parenthesis, leading to the expanded form: \( 3x^3 + 5x^2 \). This method allows you to find a single expression that represents the product of two functions at any point \( x \). It can be visualized as layering the effects of \( f(x) \) and \( g(x) \) together.

Polynomial Functions

Polynomial functions are the backbone of algebra and play a crucial part in calculus too. These functions are composed of terms consisting of variables raised to whole number powers, with coefficients. A polynomial of degree \( n \) takes the form: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \).Let’s look at our derived function from the multiplication of \( f(x) \) and \( g(x) \):

• Start with \( (f \cdot g)(x) = 3x^3 + 5x^2 \).
• This is a polynomial of degree 3 because the highest exponent is 3.
• The coefficients are 3 for \( x^3 \) and 5 for \( x^2 \), and there is no constant term or linear term here.

Polynomial functions are continuous and smooth, which means they can model various situations, from simple ones like projectile motion to more complex financial trends. Understanding how to interpret and manipulate these functions is vital for further mathematical studies.

Composite Functions

Composite functions might involve combining two functions into a single output through a function application - either sequential or through direct composition. Though we haven’t used composite functions directly in the multiplication of functions, understanding them is crucial when dealing with complex function operations.Composite functions are written as \((f \circ g)(x)\), which means you apply \( g(x) \), and then apply \( f \) to the result of \( g(x) \). For example, if you had functions \( h(x)=3x+1 \) and \( i(x) = x^2 \), then:

• The composite function \((h \circ i)(x)\) would be found by first calculating \( i(x) = x^2 \), and then applying \( h \) to that result: \( h(x^2) = 3(x^2) + 1 = 3x^2 + 1 \).
• This is different from multiplying functions, as the structure of combining them is through sequential application, not mere multiplication.

Though not directly used here, both approaches are essential tools, particularly in calculus, for understanding complex relationships in functions.