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Chapter 5: Problem 103

Given \(f(x)=2 x-3\) and \(g(x)=3 x-1,\) find the following. $$ (f \cdot g)(1) $$

Short Answer

Expert verified
The value of \((f \cdot g)(1)\) is \(-2\).

Step by step solution

01

Understand the Problem

We need to find the value of \((f \cdot g)(1)\), which means we need to multiply the functions \(f(x)\) and \(g(x)\), and then evaluate the result at \(x = 1\).
02

Define the Functions

The functions are given as \(f(x) = 2x - 3\) and \(g(x) = 3x - 1\). First, we need to multiply these two functions together.
03

Multiply the Functions

To find \((f \cdot g)(x)\), multiply the two functions:\[(f \cdot g)(x) = (2x - 3) \cdot (3x - 1)\]Use the distributive property to expand this product:\[(f \cdot g)(x) = 2x \cdot 3x + 2x \cdot -1 - 3 \cdot 3x - 3 \cdot -1\]\[(f \cdot g)(x) = 6x^2 - 2x - 9x + 3\]Combine like terms:\[(f \cdot g)(x) = 6x^2 - 11x + 3\]
04

Evaluate at x = 1

Now, substitute \(x = 1\) into the function \((f \cdot g)(x) = 6x^2 - 11x + 3\):\[(f \cdot g)(1) = 6(1)^2 - 11(1) + 3\]\[(f \cdot g)(1) = 6 - 11 + 3\]\[(f \cdot g)(1) = -2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When multiplying two functions, we often use the distributive property to simplify the process. This property helps us multiply each term in one function by every term in the other function, similar to distributing each item in a basket to buckets.
For example, if you have:
  • Function 1: \(f(x) = 2x - 3\)
  • Function 2: \(g(x) = 3x - 1\)
You can use the distributive property to find their product:
  1. Start by multiplying the first term of the first function by each term of the second function.
  2. Next, do the same with the second term of the first function.
This approach helps break down and manage more complex expressions. It鈥檚 like organizing your work into smaller, more manageable tasks.
Function Evaluation
Function evaluation is finding the output of a function for a specific input. It鈥檚 like feeding a number into a machine and watching the result come out.
In our example, we combined two functions by multiplication, and created a new polynomial, \((f \cdot g)(x) = 6x^2 - 11x + 3\).
To evaluate this function at a particular point, say \(x = 1\), we simply substitute \(1\) into every \(x\) in the expression:
  1. Calculate \(6(1)^2\).
  2. Subtract \(11(1)\).
  3. Add \(3\).
The result is \((f \, \cdot \, g)(1) = -2\).
This operation helps us understand the behavior of our function at specific values.
Polynomial Expansion
Polynomial expansion is the process of multiplying polynomials to form a single expression. This involves combining all the terms and simplifying them by collecting like terms.
For the functions \(f(x) = 2x - 3\) and \(g(x) = 3x - 1\):
  • The product is \((2x - 3)(3x - 1)\).
  • Using the distributive property, we expand this to get \(6x^2 - 2x - 9x + 3\).
After the multiplication, you collect the like terms \(-2x\) and \(-9x\), and combine them to simplify the expression to \(6x^2 - 11x + 3\).
This skill is key in algebra, allowing us to work with more complex equations in a simplified manner.

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