Let \[$ G \$\] Be The Function Given By:$\[ g(x) = \begin{cases} 3x^2 - 5x + 1 & \text{if } X \leq 0 \\ \frac{1}{x(x+1)} & \text{if } X \ \textgreater \ 0 \end{cases} \\]Which Of The Following Statements Is True?A. \[$ G

$$

Introduction

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The function $g(x)$ given by the piecewise definition is a classic example of such a function. In this article, we will analyze the function $g(x)$ and discuss its properties, including its domain, range, and behavior at different intervals.

The Piecewise Function $g(x)$

The function $g(x)$ is defined as:

$$g(x) = \begin{cases} 3x^2 - 5x + 1 & \text{if } x \leq 0 \\ \frac{1}{x(x+1)} & \text{if } x > 0 \end{cases}$$

Domain and Range

To determine the domain and range of the function $g(x)$, we need to analyze the two sub-functions separately.

Domain of $g(x)$

The domain of the first sub-function, $3x^2 - 5x + 1$, is all real numbers, since it is a polynomial function. However, the domain of the second sub-function, $\frac{1}{x(x+1)}$, is restricted to $x > 0$, since the function is undefined when $x = 0$ or $x = -1$. Therefore, the domain of $g(x)$ is $x \leq 0$ or $x > 0$, excluding $x = 0$ and $x = -1$.

Range of $g(x)$

The range of the first sub-function, $3x^2 - 5x + 1$, is all real numbers, since it is a polynomial function. However, the range of the second sub-function, $\frac{1}{x(x+1)}$, is restricted to positive real numbers, since the function is always positive. Therefore, the range of $g(x)$ is all real numbers, excluding non-positive real numbers.

Behavior at Different Intervals

To analyze the behavior of the function $g(x)$ at different intervals, we need to examine the two sub-functions separately.

Behavior at $x \leq 0$

At $x \leq 0$, the function $g(x)$ is defined by the sub-function $3x^2 - 5x + 1$. This is a quadratic function, which has a parabolic shape. The function has a minimum value at $x = -\frac{5}{6}$, and the minimum value is $-\frac{1}{12}$.

Behavior at $x > 0$

At $x > 0$, the function $g(x)$ is defined by the sub-function $\frac{1}{x(x+1)}$. This is a rational function, which has a vertical asymptote at $x = 0$ and $x = -1$. The function has a horizontal asymptote at $y = 0$.

Which of the Following Statements is True?

Now that we have analyzed the function $g(x)$, we can examine the given statements and determine which one is true.

A. $g(x)$ is a continuous function.

B. $g(x)$ is a differentiable function.

C. $g(x)$ has a maximum value at $x = 0$.

D. $g(x)$ has a minimum value at $x = -\frac{5}{6}$.

E. $g(x)$ has a vertical asymptote at $x = 0$.

F. $g(x)$ has a horizontal asymptote at $y = 0$.

Conclusion

In conclusion, the function $g(x)$ is a piecewise function that is defined by two sub-functions. The function has a domain of $x \leq 0$ or $x > 0$, excluding $x = 0$ and $x = -1$. The function has a range of all real numbers, excluding non-positive real numbers. The function has a minimum value at $x = -\frac{5}{6}$, and the minimum value is $-\frac{1}{12}$. The function has a horizontal asymptote at $y = 0$.

Answer

The correct answer is D. $g(x)$ has a minimum value at $x = -\frac{5}{6}$.

References

• [1] "Piecewise Functions." Math Open Reference, mathopenref.com/piecewise.html.
• [2] "Rational Functions." Math Open Reference, mathopenref.com/rational.html.
• [3] "Quadratic Functions." Math Open Reference, mathopenref.com/quadratic.html.
  Q&A: Let $g(x)$ be the Function Given by a Piecewise Definition ====================================================================

Q: What is the domain of the function $g(x)$?

A: The domain of the function $g(x)$ is $x \leq 0$ or $x > 0$, excluding $x = 0$ and $x = -1$.

Q: What is the range of the function $g(x)$?

A: The range of the function $g(x)$ is all real numbers, excluding non-positive real numbers.

Q: What is the behavior of the function $g(x)$ at $x \leq 0$?

A: At $x \leq 0$, the function $g(x)$ is defined by the sub-function $3x^2 - 5x + 1$. This is a quadratic function, which has a parabolic shape. The function has a minimum value at $x = -\frac{5}{6}$, and the minimum value is $-\frac{1}{12}$.

Q: What is the behavior of the function $g(x)$ at $x > 0$?

A: At $x > 0$, the function $g(x)$ is defined by the sub-function $\frac{1}{x(x+1)}$. This is a rational function, which has a vertical asymptote at $x = 0$ and $x = -1$. The function has a horizontal asymptote at $y = 0$.

Q: Is the function $g(x)$ continuous?

A: Yes, the function $g(x)$ is continuous, since it is defined by two sub-functions that are continuous at the point of intersection.

Q: Is the function $g(x)$ differentiable?

A: No, the function $g(x)$ is not differentiable at the point of intersection, since the two sub-functions have different derivatives at that point.

Q: Does the function $g(x)$ have a maximum value at $x = 0$?

A: No, the function $g(x)$ does not have a maximum value at $x = 0$, since the function is undefined at that point.

Q: Does the function $g(x)$ have a minimum value at $x = -\frac{5}{6}$?

A: Yes, the function $g(x)$ has a minimum value at $x = -\frac{5}{6}$, and the minimum value is $-\frac{1}{12}$.

Q: Does the function $g(x)$ have a vertical asymptote at $x = 0$?

A: Yes, the function $g(x)$ has a vertical asymptote at $x = 0$, since the sub-function $\frac{1}{x(x+1)}$ is undefined at that point.

Q: Does the function $g(x)$ have a horizontal asymptote at $y = 0$?

A: Yes, the function $g(x)$ has a horizontal asymptote at $y = 0$, since the sub-function $\frac{1}{x(x+1)}$ approaches 0 as $x$ approaches infinity.

Conclusion

In conclusion, the function $g(x)$ is a piecewise function that is defined by two sub-functions. The function has a domain of $x \leq 0$ or $x > 0$, excluding $x = 0$ and $x = -1$. The function has a range of all real numbers, excluding non-positive real numbers. The function has a minimum value at $x = -\frac{5}{6}$, and the minimum value is $-\frac{1}{12}$. The function has a horizontal asymptote at $y = 0$.

References

• [1] "Piecewise Functions." Math Open Reference, mathopenref.com/piecewise.html.
• [2] "Rational Functions." Math Open Reference, mathopenref.com/rational.html.
• [3] "Quadratic Functions." Math Open Reference, mathopenref.com/quadratic.html.