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Chapter 10: Problem 61

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=-3 x+1 $$

Short Answer

Expert verified
The function is decreasing on \((-\infty, \infty)\).

Step by step solution

01

Identify the Function Type

The given function is a linear function of the form \( f(x) = ax + b \), where \( a = -3 \). This implies it is a straight line.
02

Determine the Slope

The slope of the function \( f(x) = -3x + 1 \) is \( -3 \). Since the slope is negative, the line decreases as \( x \) increases.
03

Establish Increasing or Decreasing Nature

For linear functions, the sign of the slope determines whether the function is increasing or decreasing. A negative slope (\( -3 \) in this case) indicates that the function is decreasing over its entire domain.
04

Define the Domain

The domain of a linear function like \( f(x) = -3x + 1 \) is all real numbers, \( (-\infty, \infty) \).
05

State the Intervals

Since the slope is negative, \( f(x) \) is decreasing on \( (-\infty, \infty) \) and there are no intervals where it is increasing.

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Key Concepts

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

Slope Determination
In a linear function, the slope is a crucial concept that tells us how steep the line is and its direction. The slope, often represented by "a" in the linear equation form \( f(x) = ax + b \), is a number that can be positive, negative, or zero. It fundamentally describes how the function behaves and changes as \( x \) changes.

To determine the slope, take a look at the coefficient of \( x \) in the equation. For example, in the linear function \( f(x) = -3x + 1 \), the coefficient of \( x \) is \(-3\), which is the slope.

  • If the slope is positive, the line ascends as it moves from left to right on the graph, indicating the function is increasing.
  • If the slope is zero, the line is flat and the function is constant.
  • If the slope is negative, like \(-3\), the line descends, meaning the function decreases as \( x \) gets larger.
Recognizing the slope's sign is key to understanding the linear function's behavior.
Increasing and Decreasing Intervals
Understanding where a function is increasing or decreasing is important for grasping its overall behavior. When dealing with linear functions, this determination becomes straightforward.

For the function \( f(x) = -3x + 1 \), the fact that the slope is negative automatically tells us that the function decreases over its entire range of \( x \)-values. This is because, as \( x \) increases, \( f(x) \) will always decrease when the slope is negative. To put it simply,
  • Due to the negative slope of \( -3 \), \( f(x) \) never increases - it's always going downhill.
  • The entire domain, or all possible \( x \) values, is part of this decreasing interval.
So, for this linear function, we can say it is decreasing for all \( x \) in the interval \((-\infty, \infty)\). There are no increasing intervals here.
Domain of a Function
The domain of a function refers to all the possible input values, typically represented by \( x \), that will yield a valid output, \( f(x) \). Understanding the domain is critical in identifying which \( x \)-values a function can accept without any restrictions.

For a linear function like \( f(x) = -3x + 1 \), the domain is straightforward and encompasses all real numbers, denoted as \((-\infty, \infty)\). This means any real number substituted for \( x \) will result in a valid output.

  • There are no restrictions like those found in rational functions, where division by zero is a concern.
  • No square roots leading to the need for non-negative inside values.
Thus, understanding the domain helps us see that any point on the line of a linear function is valid, showing its applicability across the entire number line without interruption.

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