Evaluating the Function on Different Intervals

Oct 13, 2025 by windailys.com 97 views

Introduction

When dealing with polynomial functions, understanding the behavior of the function on different intervals is crucial. In this case, we are given the function $f(x)=x(x+3)(x+1)(x-4)$ and asked to determine the sign of $f$ on the interval $-1\ \textless \ x\ \textless \ 4$ . To approach this problem, we need to analyze the function's behavior on the given interval, taking into account the function's zeros and the behavior of the function as $x$ approaches the endpoints of the interval.

Understanding the Function's Zeros

The function $f(x)=x(x+3)(x+1)(x-4)$ has zeros at $x=-3, x=-1, x=0$ , and $x=4$ . This means that the function intersects the x-axis at these points, resulting in a change in the function's sign. To understand the sign of $f$ on the interval $-1\ \textless \ x\ \textless \ 4$ , we need to examine the behavior of the function between these zeros.

Analyzing the Function's Behavior on the Interval

To determine the sign of $f$ on the interval $-1\ \textless \ x\ \textless \ 4$ , we can use the concept of sign charts. A sign chart is a graphical representation of the sign of a function on different intervals. By examining the sign chart, we can determine the sign of the function on the given interval.

Creating a Sign Chart

To create a sign chart, we need to examine the sign of the function on different intervals. We can do this by choosing test points within each interval and evaluating the function at these points. The sign chart will help us visualize the behavior of the function on the given interval.

Evaluating the Function on Different Intervals

Let's evaluate the function on different intervals to create a sign chart.

Interval $(-\infty, -3)$ : On this interval, all the factors of the function are negative, resulting in a negative value for the function.
Interval $(-3, -1)$ : On this interval, the factor $(x+3)$ is positive, while the other factors are negative. This results in a positive value for the function.
Interval $(-1, 0)$ : On this interval, the factors $(x+3)$ and $(x+1)$ are positive, while the factors $(x)$ and $(x-4)$ are negative. This results in a negative value for the function.
Interval $(0, 4)$ : On this interval, the factors $(x+3)$ , $(x+1)$ , and $(x)$ are positive, while the factor $(x-4)$ is negative. This results in a negative value for the function.

Determining the Sign of $f$ on the Interval $-1\ \textless \ x\ \textless \ 4$

Based on the sign chart, we can determine the sign of $f$ on the interval $-1\ \textless \ x\ \textless \ 4$ . Since the function is negative on this interval, the correct answer is:

A. $f$ is always negative on the interval $-1\ \textless \ x\ \textless \ 4$

Conclusion

In conclusion, the function $f(x)=x(x+3)(x+1)(x-4)$ has zeros at $x=-3, x=-1, x=0$ , and $x=4$ . By analyzing the function's behavior on the interval $-1\ \textless \ x\ \textless \ 4$ , we can determine that the function is always negative on this interval. This is a crucial concept in mathematics, as understanding the behavior of functions on different intervals is essential in solving problems and making predictions.

Frequently Asked Questions

What is the sign of $f$ on the interval $-1\ \textless \ x\ \textless \ 4$ ?
- The function $f$ is always negative on the interval $-1\ \textless \ x\ \textless \ 4$ .
What are the zeros of the function $f(x)=x(x+3)(x+1)(x-4)$ ?
- The zeros of the function are $x=-3, x=-1, x=0$ , and $x=4$ .
How can we determine the sign of a function on a given interval?
- We can use a sign chart to determine the sign of a function on a given interval. A sign chart is a graphical representation of the sign of a function on different intervals.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we explored the function $f(x)=x(x+3)(x+1)(x-4)$ and determined that the function is always negative on the interval $-1\ \textless \ x\ \textless \ 4$ . In this article, we will address some of the most frequently asked questions related to this function.

Q&A

Q: What is the sign of $f$ on the interval $-1\ \textless \ x\ \textless \ 4$ ?

A: The function $f$ is always negative on the interval $-1\ \textless \ x\ \textless \ 4$ .

Q: What are the zeros of the function $f(x)=x(x+3)(x+1)(x-4)$ ?

A: The zeros of the function are $x=-3, x=-1, x=0$ , and $x=4$ .

Q: How can we determine the sign of a function on a given interval?

A: We can use a sign chart to determine the sign of a function on a given interval. A sign chart is a graphical representation of the sign of a function on different intervals.

Q: What is the behavior of the function $f(x)=x(x+3)(x+1)(x-4)$ as $x$ approaches the endpoints of the interval $-1\ \textless \ x\ \textless \ 4$ ?

A: As $x$ approaches $-1$ from the left, the function $f(x)$ approaches $0$ from the negative side. As $x$ approaches $-1$ from the right, the function $f(x)$ approaches $0$ from the positive side. As $x$ approaches $4$ from the left, the function $f(x)$ approaches $0$ from the negative side. As $x$ approaches $4$ from the right, the function $f(x)$ approaches $0$ from the positive side.

Q: Can we use the concept of sign charts to determine the sign of a function on an interval that contains a zero of the function?

A: Yes, we can use the concept of sign charts to determine the sign of a function on an interval that contains a zero of the function. However, we need to be careful when choosing test points, as the function may change sign at the zero.

Q: How can we use the concept of sign charts to determine the sign of a function on an interval that contains multiple zeros of the function?

A: We can use the concept of sign charts to determine the sign of a function on an interval that contains multiple zeros of the function by choosing test points that are not equal to the zeros of the function. This will help us to determine the sign of the function on the interval.

Conclusion

In conclusion, the function $f(x)=x(x+3)(x+1)(x-4)$ has zeros at $x=-3, x=-1, x=0$ , and $x=4$ . By analyzing the function's behavior on the interval $-1\ \textless \ x\ \textless \ 4$ , we can determine that the function is always negative on this interval. We can also use the concept of sign charts to determine the sign of a function on a given interval, including intervals that contain zeros of the function.

Frequently Asked Questions: Additional Resources

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline
[4] "Sign Charts: A Graphical Approach to Determining the Sign of a Function" by [Author's Name]

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline
[4] "Sign Charts: A Graphical Approach to Determining the Sign of a Function" by [Author's Name]

[1] "The Function $f(x)=x(x+3)(x+1)(x-4)$ : An Introduction"
[2] "Determining the Sign of a Function on a Given Interval"
[3] "Sign Charts: A Graphical Approach to Determining the Sign of a Function"

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