Problem 63 - Precalculus : Building Concepts and Connections

Question

Applications In this set of exercises, you will use properties of functions to study real-world problems. Revenue The revenue for a company is given by \(R(x)=30 x\) where \(x\) is the number of units sold in thousands. Is this an increasing or a decreasing function? Explain.

Step-by-Step Solution

Answer

The function \(R(x)=30x\) is an increasing function because its leading coefficient is positive.

Step 1: Identify the nature of the function

The function \(R(x)=30x\) is a linear function, which is a first degree polynomial.

Step 2: Identify the leading coefficient

The leading coefficient of a linear function is the number that multiplies the variable. In the function \(R(x)=30x\), the leading coefficient is 30.

Step 3: Determine whether the function is increasing or decreasing

Because the leading coefficient (30) is positive, the function \(R(x)=30x\) is an increasing function. This means that as the value of \(x\) (the number of units sold) increases, the value of \(R(x)\) (the company's revenue) also increases.

Key Concepts

Linear Functions

Linear functions are among the simplest and most fundamental types of functions in mathematics. A linear function can be expressed in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This creates a straight line on a graph, hence the name 'linear'.
These functions are characterized by:

• A constant rate of change: The slope \(m\) indicates how much \(f(x)\) changes for each unit increase in \(x\).
  
• No curves: The graph of a linear function is always a straight line.
  
• Simple equations: Linear functions involve only the first power of \(x\).
  

In the revenue example \(R(x) = 30x\), the function is linear because it fits the standard form with \(m = 30\) and \(b = 0\). This is indicative of a direct proportion where revenue grows linearly as more units are sold.

Increasing Functions

An increasing function is a function where, as the input \(x\) increases, the output \(f(x)\) also increases. This is a key concept when you want to determine if a function is growing or contracting.
Key points to identify an increasing function:

• The slope or derivative \(f'(x)\) is positive: This shows an upward trend.
  
• The graph ascends from left to right: This is the typical appearance of an increasing function.
  

In our revenue function example \(R(x) = 30x\), the slope (or leading coefficient) is 30, which is positive. Thus, the function is increasing. This means that as the company sells more units, the revenue grows accordingly, providing a straightforward model of growth.

Polynomials

Polynomials are expressions comprising variables and coefficients, connected by operations of addition, subtraction, multiplication, and non-negative integer exponents. They form the basis for various functions, including linear, quadratic, cubic, and higher-degree functions.
Important features to remember about polynomials:

• Degree: This is the highest power of the variable. It determines the polynomial's overall shape and classification.
  
• Leading coefficient: The coefficient of the term with the highest degree. It influences the function's growth rate and direction.
  

In the context of our example, \(R(x) = 30x\) is a first-degree polynomial, meaning it is linear. The simplicity of linear polynomials makes them useful in modeling straightforward relationships, like the direct correlation between sales and revenue in our example.