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Chapter 2: Problem 8

Determine if each function is increasing or decreasing $$ g(x)=5 x+6 $$

Short Answer

Expert verified
The function is increasing.

Step by step solution

01

Identify the type of function

The given function is \( g(x) = 5x + 6 \), which is a linear function. Linear functions are of the form \( f(x) = ax + b \).
02

Determine the slope

In the linear function \( g(x) = 5x + 6 \), the coefficient of \( x \) is the slope, which is \( a = 5 \).
03

Analyze the slope

If the slope \( a > 0 \), then the function is increasing. If \( a < 0 \), the function is decreasing. Here, since \( a = 5 \) and \( 5 > 0 \), the function is increasing.

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Key Concepts

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

Increasing Functions
In the world of linear functions, an increasing function is one where, as we move from left to right along the graph, the function values become larger. This means that as the input (usually referred to as 'x') increases, the output also increases.

For linear functions such as \( g(x) = 5x + 6 \), the determining factor for whether the function is increasing lies in the slope. The slope is the number that is multiplied by 'x' in the function equation. In this example, the slope is 5, which is positive.

  • If the slope is greater than zero (\( a > 0 \)), the function increases as you move along the x-axis to the right.
  • An increasing function will rise upwards from left to right on the graph.
Thus, a positive slope directly results in an increasing function, giving us a positive rate of change as we move along the x-axis.
Decreasing Functions
Decreasing functions are essentially the opposite of increasing functions. For a function to be decreasing, as you move along the x-axis to the right, the values of the function decrease. This means output becomes smaller as input grows.

While our original example \( g(x) = 5x + 6 \) does not exhibit decreasing behavior (due to its positive slope), it's still important to understand what characterizes decreasing functions.

  • If a linear function has a negative slope (\( a < 0 \)), it is decreasing.
  • Graphically, a decreasing function will slope downwards from left to right.
When handling real-world problems or different kinds of functions, recognizing a negative slope will alert you that the function is decreasing, reflecting a descent as you move across the graph.
Slope Analysis
Slope is a key concept in understanding how linear functions behave. Simply put, the slope is a measure of how steep a line is and the direction it moves. In the function \( g(x) = 5x + 6 \), the slope is represented by the number in front of 'x', which is 5. Here鈥檚 how you can think about slope:

  • Slope tells us how much the function value (or y-value) changes for every one unit increase in the x-value.
  • A positive slope means the function ascends as it moves from left to right. For our function \( g(x) = 5x + 6 \), the slope of 5 suggests for every unit increase in 'x', the function value increases by 5 units.
  • Conversely, a negative slope means the function descends.
Understanding slope is crucial, not just for determining if a function is increasing or decreasing, but for interpreting those rates of changes in relation to real-world phenomena, such as speed, economics, and other disciplines where change matters.

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Most popular questions from this chapter

A farmer finds there is a linear relationship between the number of bean stalks, \(n,\) she plants and the yield, \(y\), each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form \(y=m n+b\) that gives the yield when \(n\) stalks are planted.

Given each set of information, find a linear equation satisfying the conditions, if possible $$ f(-1)=4, \text { and } f(5)=1 $$

Determine if each function is increasing or decreasing $$ j(x)=\frac{1}{2} x-3 $$

Find the horizontal and vertical intercepts of each equation. $$ f(x)=-x+2 $$

Find the area of a parallelogram bounded by the \(x\) axis, the line \(g(x)=2,\) the line \(f(x)=3 x,\) and the line parallel to \(f(x)\) passing through (6,1)
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