Understanding Decreasing Functions

Oct 13, 2025 by wbrum.org 27 views

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function can be increasing, decreasing, or constant over a given interval. In this article, we will focus on identifying a function that is decreasing only over the interval $(-1, \infty)$ .

Understanding Decreasing Functions

A decreasing function is a function that takes on smaller and smaller values as the input increases. In other words, as the input $x$ increases, the output $f(x)$ decreases. A function can be decreasing over a specific interval, and it can be increasing or constant over other intervals.

Analyzing the Given Table

The given table shows the values of a function $f(x)$ for different values of $x$ . We need to analyze the table to determine which function is decreasing only over the interval $(-1, \infty)$ .

$x$	$f(x)$
-3	-1
-2	-3
-1	-5
0	-2
1	-1
2

Examining the Function Values

Let's examine the function values in the table to determine which function is decreasing only over the interval $(-1, \infty)$ .

For $x = -3$ , $f(x) = -1$ .
For $x = -2$ , $f(x) = -3$ .
For $x = -1$ , $f(x) = -5$ .
For $x = 0$ , $f(x) = -2$ .
For $x = 1$ , $f(x) = -1$ .
For $x = 2$ , $f(x)$ is not given.

Identifying the Decreasing Function

From the table, we can see that the function values are decreasing as $x$ increases from $-3$ to $-1$ . However, the function value at $x = 0$ is $-2$ , which is greater than the function value at $x = -1$ . This indicates that the function is not decreasing over the entire interval $(-1, \infty)$ .

However, if we consider the function values for $x = -3$ , $-2$ , and $-1$ , we can see that the function is decreasing over this interval. Therefore, the function that is decreasing only over the interval $(-1, \infty)$ is the function with the values $f(-3) = -1$ , $f(-2) = -3$ , and $f(-1) = -5$ .

Conclusion

In conclusion, the function that is decreasing only over the interval $(-1, \infty)$ is the function with the values $f(-3) = -1$ , $f(-2) = -3$ , and $f(-1) = -5$ . This function is decreasing over the interval $(-1, \infty)$ , but it is not decreasing over the entire interval $(-\infty, \infty)$ .

Discussion

The concept of decreasing functions is an important topic in mathematics, particularly in calculus. Understanding decreasing functions is crucial in solving optimization problems, finding maximum and minimum values, and analyzing the behavior of functions.

In this article, we analyzed a table of function values to identify the function that is decreasing only over the interval $(-1, \infty)$ . We examined the function values in the table and determined that the function with the values $f(-3) = -1$ , $f(-2) = -3$ , and $f(-1) = -5$ is the function that is decreasing only over the interval $(-1, \infty)$ .

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer

Keywords

Decreasing functions
Interval
Function values
Calculus
Optimization problems
Maximum and minimum values
Function behavior
Frequently Asked Questions (FAQs) about Decreasing Functions ================================================================

Q: What is a decreasing function?

A: A decreasing function is a function that takes on smaller and smaller values as the input increases. In other words, as the input $x$ increases, the output $f(x)$ decreases.

Q: How do I determine if a function is decreasing?

A: To determine if a function is decreasing, you need to examine the function values over a given interval. If the function values are getting smaller and smaller as the input increases, then the function is decreasing over that interval.

Q: What is the difference between a decreasing function and a constant function?

A: A decreasing function takes on smaller and smaller values as the input increases, while a constant function takes on the same value for all inputs. In other words, a decreasing function has a negative slope, while a constant function has a slope of zero.

Q: Can a function be both increasing and decreasing over different intervals?

A: Yes, a function can be both increasing and decreasing over different intervals. For example, a function can be increasing over the interval $(-\infty, -1)$ and decreasing over the interval $(-1, \infty)$ .

Q: How do I find the maximum and minimum values of a decreasing function?

A: To find the maximum and minimum values of a decreasing function, you need to examine the function values over the given interval. The maximum value will be the largest function value, while the minimum value will be the smallest function value.

Q: Can a decreasing function have a maximum value?

A: No, a decreasing function cannot have a maximum value. By definition, a decreasing function takes on smaller and smaller values as the input increases, so it will never have a maximum value.

Q: Can a decreasing function have a minimum value?

A: Yes, a decreasing function can have a minimum value. The minimum value will be the smallest function value over the given interval.

Q: How do I graph a decreasing function?

A: To graph a decreasing function, you need to plot the function values over the given interval. The graph will show the function values getting smaller and smaller as the input increases.

Q: Can a decreasing function be represented by a linear equation?

A: Yes, a decreasing function can be represented by a linear equation. For example, the function $f(x) = -2x + 1$ is a decreasing function that can be represented by a linear equation.

Q: Can a decreasing function be represented by a quadratic equation?

A: Yes, a decreasing function can be represented by a quadratic equation. For example, the function $f(x) = -x^2 + 2x + 1$ is a decreasing function that can be represented by a quadratic equation.

Q: Can a decreasing function be represented by a polynomial equation of degree 3 or higher?

A: Yes, a decreasing function can be represented by a polynomial equation of degree 3 or higher. For example, the function $f(x) = -x^3 + 2x^2 + x + 1$ is a decreasing function that can be represented by a polynomial equation of degree 3.

Q: Can a decreasing function be represented by a rational function?

A: Yes, a decreasing function can be represented by a rational function. For example, the function $f(x) = -\frac{1}{x + 1}$ is a decreasing function that can be represented by a rational function.

Q: Can a decreasing function be represented by a trigonometric function?

A: Yes, a decreasing function can be represented by a trigonometric function. For example, the function $f(x) = -\sin(x)$ is a decreasing function that can be represented by a trigonometric function.

Q: Can a decreasing function be represented by an exponential function?

A: Yes, a decreasing function can be represented by an exponential function. For example, the function $f(x) = -e^x$ is a decreasing function that can be represented by an exponential function.

Q: Can a decreasing function be represented by a logarithmic function?

A: Yes, a decreasing function can be represented by a logarithmic function. For example, the function $f(x) = -\log(x)$ is a decreasing function that can be represented by a logarithmic function.

Conclusion

In conclusion, decreasing functions are an important topic in mathematics, particularly in calculus. Understanding decreasing functions is crucial in solving optimization problems, finding maximum and minimum values, and analyzing the behavior of functions. We hope that this FAQ article has provided you with a better understanding of decreasing functions and how to work with them.