Problem 68 - Algebra and Trigonometry

Question

Given \(f(x)=6 x+5\) and \(g(x)=x^{2}-3 x+2,\) find each of the following: a. \((f \circ g)(x)\) b. \((g \cdot f)(x)\) c. \((f \circ g)(-1)\)

Step-by-Step Solution

Answer

The solutions are: a. The composed function \( (f \circ g)(x) = 6x^{2}-18x+17 \), b. The multiplied function \( (g \cdot f)(x) = 6x^{3}-13x^{2}-3x+10 \), and c. The value of the composed function at -1 is \( (f \circ g)(-1) = 41 \).

Step 1: Find (f ∘ g)(x)

Function composition \( (f \circ g)(x) \) means we plug g(x) into f(x). That means we substitute \( x^{2}-3 x+2 \) into \( 6x+5 \). That gives us \( f(g(x)) = 6(g(x)) + 5 = 6(x^{2}-3 x+2) + 5 = 6x^{2}-18x+12 + 5 = 6x^{2}-18x+17 \).

Step 2: Find (g ⋅ f)(x)

Function multiplication \( (g \cdot f)(x) \) means we multiply g(x) by f(x). So, we get \( g(x) \cdot f(x) = (x^{2}-3x+2) \cdot (6x+5) = 6x^{3} -18x^{2} + 12x +5x^{2} -15x +10 = 6x^{3}-13x^{2}-3x+10 \).

Step 3: Find (f ∘ g)(-1)

We have already calculated \( (f \circ g)(x) \) in step 1, so now plug in x = -1 into the derived function. Substitute -1 into \( 6x^{2}-18x+17 \) to get \( (f \circ g)(-1) = 6(-1)^{2}-18(-1)+17 = 6+18+17 = 41 \).

Key Concepts

Function Composition

Function composition is all about combining two functions. When you see

• \((f \circ g)(x)\)

it means you need to put one function inside another. Essentially, you take the output of one function and use it as the input for the next function!
This is like following a two-step process. Let's break it down with our given functions:- **Defined Functions**:

• \(f(x) = 6x + 5\)
• \(g(x) = x^2 - 3x + 2\)

- **Perform the Composition**:

• First, you compute \(g(x)\), which is \(x^2 -3x + 2\).
• Next, you replace every \(x\) in \(f(x)\) with \(g(x)\). So, \(f(g(x)) = 6(g(x)) + 5\).
• This turns into \(6(x^2 - 3x + 2) + 5\).
Once you simplify, you'll get the result: \(6x^2 - 18x + 17\).

This shows that the composition of functions gives you a new polynomial function!

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to positive integer powers and their coefficients. Here's a brief breakdown to help you understand them:- **Basics of Polynomial Functions**:

• A typical polynomial can look like: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants.
• Each piece, such as \(a_nx^n\), is called a term.

- **Our Example**:

• \(g(x) = x^2 - 3x + 2\) is a polynomial function of degree 2.
• \(f(x) = 6x + 5\) is a polynomial function of degree 1.

- **Combining Polynomials**:

• When functions are combined, as in \((g \cdot f)(x)\), you multiply the polynomials together.
• The result, \(6x^3 - 13x^2 - 3x + 10\), is another polynomial function, now of degree 3.

Polynomial functions are incredibly versatile and serve as building blocks for many more complex functions you'll encounter.

Substitution Method

Substitution is a straightforward but crucial method in mathematics used to simplify expressions or solve equations. When working with function operations, substitution comes handy in two primary ways:- **Within Functions**:

• To substitute, simply replace the variable (like \(x\)) in a function by another expression or a specific number.
• In the solution, we used substitution to find \((f \circ g)(-1)\).
• This meant taking the result of \((f \circ g)(x)\) and plugging \(x = -1\) into it.
- **Our Calculation**:
• Start with the derived function \(6x^2 - 18x + 17\).
• Substitute \(-1\) for \(x\): \(6(-1)^2 - 18(-1) + 17\).
• By solving this, you find the value: \(41\).


Substitution is an essential method to know, allowing you to evaluate the values of composed functions and approach more complex problems confidently.