Problem 65 - Brief Applied Calculus

Question

6.5-78. Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ f(x)=7 x-5 $$

Step-by-Step Solution

Answer

The function \( f(x) = 7x - 5 \) is continuous everywhere.

Step 1: Understanding the Function

The function given is a linear function, which has the form \( f(x) = mx + b \). In this case, \( f(x) = 7x - 5 \) where \( m = 7 \) and \( b = -5 \). Linear functions are continuous everywhere on their domain, which is all real numbers.

Step 2: Checking for Continuity

Since \( f(x) = 7x - 5 \) is a linear function, it is continuous for all real values of \( x \). Linear functions do not have any breaks, holes, or jumps in their graphs, indicating they are continuous everywhere.

Key Concepts

Linear Functions

Linear functions are a fundamental concept in mathematics. A linear function has the general form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. Here, \( m \) is the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.
Linear functions represent the simplest form of a function, as they create a straight line when graphed.

• They have a constant rate of change, meaning the slope \( m \) remains the same throughout.
• The output (\( f(x) \)) changes the same amount for a constant change in input (\( x \)).

Linear functions are widely used because of their simplicity and predictability. They are deterministic; given an input, they always produce the same output.
In the context of the exercise, the function \( f(x) = 7x - 5 \) follows this linear form, with a slope of 7 and a y-intercept of -5.

Continuous Functions

Continuity in functions is an important concept in calculus. A function is considered continuous at a point if there are no breaks or interruptions at that point.
A continuous function is one you can draw without lifting your pencil from the paper.The graph of a continuous function will have no:

• Gaps: points where the function suddenly stops and restarts.
• Holes: undefined points in the function where there might be a limit, but no value.
• Jumps: sudden changes in function value as \( x \) moves through the domain.

Linear functions, such as \( f(x) = 7x - 5 \), are always continuous across their domain of all real numbers, as they don't have discontinuities by their nature.
This means you can expect a smooth and unbroken line when graphing a linear function.

Real Numbers

Real numbers form a fundamental building block in mathematics. They include both rational numbers (such as fractions \( \frac{2}{3} \), and whole numbers) and irrational numbers (numbers that cannot be expressed as a fraction, like \( \pi \) and \( \sqrt{2} \)).
Real numbers are:

• Complete: they include all numbers that can be found on the number line, covering every point from negative infinity to positive infinity.
• Ordered: any two real numbers can be compared, determining which is larger or ordering several from smallest to largest.

In the context of the given function \( f(x) = 7x - 5 \), the domain is all real numbers. This means \( x \) can take any real number value, from the smallest to the largest, and \( f(x) \) will remain defined and continuous.
Understanding real numbers is crucial for grasping more complex mathematical topics, as they provide a base for further mathematical exploration and application.