Comparing the Intervals Where Both Functions are Negative

Oct 13, 2025 by thesweetdesigns.com 3 views

When Comparing the Functions $f(x)=x^2+2x$ and $g(x)=\log(2x+1)$ : Finding the Interval Where Both Functions are Negative

When comparing two functions, it's essential to understand their behavior and how they interact with each other. In this case, we're given two functions: $f(x)=x^2+2x$ and $g(x)=\log(2x+1)$ . Our goal is to find the interval where both functions are negative. To do this, we'll need to analyze the behavior of each function separately and then compare their values.

Analyzing the Function $f(x)=x^2+2x$

The function $f(x)=x^2+2x$ is a quadratic function, which means it has a parabolic shape. To find the interval where $f(x)$ is negative, we need to determine the values of $x$ for which the function is below the x-axis.

Finding the Roots of $f(x)$

The roots of a quadratic function are the values of $x$ where the function intersects the x-axis. To find the roots of $f(x)$ , we can set the function equal to zero and solve for $x$ .

x^2+2x=0

Factoring out $x$ , we get:

x(x+2)=0

This tells us that either $x=0$ or $x=-2$ . These are the roots of the function $f(x)$ .

Determining the Interval Where $f(x)$ is Negative

Now that we have the roots of $f(x)$ , we can determine the interval where the function is negative. Since the parabola opens upwards, the function is negative between the roots.

-2<x<0

This is the interval where $f(x)$ is negative.

Analyzing the Function $g(x)=\log(2x+1)$

The function $g(x)=\log(2x+1)$ is a logarithmic function. To find the interval where $g(x)$ is negative, we need to determine the values of $x$ for which the function is below the x-axis.

Finding the Domain of $g(x)$

The domain of a logarithmic function is the set of all real numbers for which the argument (the expression inside the logarithm) is positive. In this case, the argument is $2x+1$ , which must be greater than zero.

2x+1>0

Solving for $x$ , we get:

x>-1/2

This is the domain of the function $g(x)$ .

Determining the Interval Where $g(x)$ is Negative

Now that we have the domain of $g(x)$ , we can determine the interval where the function is negative. Since the logarithmic function is decreasing, the function is negative when the argument is less than 1.

2x+1<1

Solving for $x$ , we get:

x<0

However, we also need to consider the domain of the function, which is $x>-1/2$ . Therefore, the interval where $g(x)$ is negative is:

-1/2<x<0

Comparing the Intervals Where Both Functions are Negative

Now that we have the intervals where both functions are negative, we can compare them to find the common interval.

The interval where $f(x)$ is negative is:

-2<x<0

The interval where $g(x)$ is negative is:

-1/2<x<0

The only interval that satisfies both conditions is:

-1/2<x<0

This is the interval where both functions are negative.

In conclusion, when comparing the functions $f(x)=x^2+2x$ and $g(x)=\log(2x+1)$ , we found that the interval where both functions are negative is $-1/2<x<0$ . This interval satisfies the conditions for both functions to be negative.

The correct answer is:

B. $(-1,1$ ] is incorrect as it does not satisfy the conditions for both functions to be negative.
A. $(-\infty,-2$ ] is incorrect as it does not satisfy the conditions for both functions to be negative.
D. $(-\infty, \infty$ ] is incorrect as it does not satisfy the conditions for both functions to be negative.
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When Comparing the Functions $f(x)=x^2+2x$ and $g(x)=\log(2x+1)$ : Finding the Interval Where Both Functions are Negative

Q: What is the main goal of this article?

A: The main goal of this article is to find the interval where both functions $f(x)=x^2+2x$ and $g(x)=\log(2x+1)$ are negative.

Q: What are the two functions being compared in this article?

A: The two functions being compared in this article are $f(x)=x^2+2x$ and $g(x)=\log(2x+1)$ .

Q: What is the first step in finding the interval where both functions are negative?

A: The first step in finding the interval where both functions are negative is to analyze the behavior of each function separately.

Q: How do we find the roots of the function $f(x)=x^2+2x$ ?

A: We find the roots of the function $f(x)=x^2+2x$ by setting the function equal to zero and solving for $x$ .

Q: What are the roots of the function $f(x)=x^2+2x$ ?

A: The roots of the function $f(x)=x^2+2x$ are $x=0$ and $x=-2$ .

Q: How do we determine the interval where $f(x)$ is negative?

A: We determine the interval where $f(x)$ is negative by finding the values of $x$ for which the function is below the x-axis.

Q: What is the interval where $f(x)$ is negative?

A: The interval where $f(x)$ is negative is $-2<x<0$ .

Q: How do we find the domain of the function $g(x)=\log(2x+1)$ ?

A: We find the domain of the function $g(x)=\log(2x+1)$ by determining the values of $x$ for which the argument (the expression inside the logarithm) is positive.

Q: What is the domain of the function $g(x)=\log(2x+1)$ ?

A: The domain of the function $g(x)=\log(2x+1)$ is $x>-1/2$ .

Q: How do we determine the interval where $g(x)$ is negative?

A: We determine the interval where $g(x)$ is negative by finding the values of $x$ for which the function is below the x-axis.

Q: What is the interval where $g(x)$ is negative?

A: The interval where $g(x)$ is negative is $-1/2<x<0$ .

Q: How do we compare the intervals where both functions are negative?

A: We compare the intervals where both functions are negative by finding the common interval that satisfies both conditions.

Q: What is the interval where both functions are negative?

A: The interval where both functions are negative is $-1/2<x<0$ .

Q: What is the main difference between the two functions?

A: The main difference between the two functions is that $f(x)=x^2+2x$ is a quadratic function, while $g(x)=\log(2x+1)$ is a logarithmic function.

Q: How do we find the roots of a quadratic function?

A: We find the roots of a quadratic function by setting the function equal to zero and solving for $x$ .

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is the set of all real numbers for which the argument (the expression inside the logarithm) is positive.

Q: How do we determine the interval where a function is negative?

A: We determine the interval where a function is negative by finding the values of $x$ for which the function is below the x-axis.

Q: What is the interval where both functions are negative?

A: The interval where both functions are negative is $-1/2<x<0$ .