What Are The $x$-intercept(s) Of The Function $f(x)=\frac{5x^2-25x}{x}$?A. $ X = 5 X=5 X = 5 [/tex]B. $x=0$ And $x=5$C. $ X = 0 X=0 X = 0 [/tex]D. $x=-5$
Understanding the Concept of $x$-Intercept
The $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis. In other words, it is the value of $x$ at which the function has a value of zero. To find the $x$-intercept of a function, we need to set the function equal to zero and solve for $x$.
Analyzing the Given Function
The given function is $f(x)=\frac{5x^2-25x}{x}$. To find the $x$-intercept of this function, we need to set it equal to zero and solve for $x$. This can be done by setting the numerator of the function equal to zero and solving for $x$.
Setting the Numerator Equal to Zero
The numerator of the function is $5x^2-25x$. To find the values of $x$ at which the function has a value of zero, we need to set this expression equal to zero and solve for $x$. This can be done by factoring the expression or by using the quadratic formula.
Factoring the Expression
The expression $5x^2-25x$ can be factored as $5x(x-5)$. Setting this expression equal to zero gives us the equation $5x(x-5)=0$. This equation can be solved by setting each factor equal to zero and solving for $x$.
Solving for $x$
Setting the first factor equal to zero gives us the equation $5x=0$. Solving for $x$ gives us $x=0$. Setting the second factor equal to zero gives us the equation $x-5=0$. Solving for $x$ gives us $x=5$.
Conclusion
The $x$-intercept(s) of the function $f(x)=\frac{5x^2-25x}{x}$ are $x=0$ and $x=5$. This can be verified by plugging these values of $x$ into the function and checking that the function has a value of zero at these points.
Final Answer
The final answer is B. $x=0$ and $x=5$.
Step-by-Step Solution
- Set the numerator of the function equal to zero: $5x^2-25x=0$
- Factor the expression: $5x(x-5)=0$
- Set each factor equal to zero and solve for $x$:
- The $x$-intercept(s) of the function are $x=0$ and $x=5$.
Common Mistakes
- Failing to set the numerator equal to zero
- Failing to factor the expression
- Failing to set each factor equal to zero and solve for $x$
- Failing to check that the function has a value of zero at the $x$-intercept(s)
Real-World Applications
- Finding the $x$-intercept(s) of a function is an important concept in mathematics and has many real-world applications, such as:
- Modeling population growth
- Modeling the spread of disease
- Modeling the motion of objects
- Understanding the concept of $x$-intercept is crucial in many fields, such as physics, engineering, and economics.
Conclusion
In conclusion, the $x$-intercept(s) of the function $f(x)=\frac{5x^2-25x}{x}$ are $x=0$ and $x=5$. This can be verified by plugging these values of $x$ into the function and checking that the function has a value of zero at these points. Understanding the concept of $x$-intercept is crucial in many fields and has many real-world applications.
Frequently Asked Questions
Q: What is the $x$-intercept of a function?
A: The $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis. In other words, it is the value of $x$ at which the function has a value of zero.
Q: How do I find the $x$-intercept of a function?
A: To find the $x$-intercept of a function, you need to set the function equal to zero and solve for $x$. This can be done by setting the numerator of the function equal to zero and solving for $x$.
Q: What is the difference between the $x$-intercept and the $y$-intercept?
A: The $x$-intercept is the point at which the graph of the function crosses the $x$-axis, while the $y$-intercept is the point at which the graph of the function crosses the $y$-axis.
Q: Can a function have more than one $x$-intercept?
A: Yes, a function can have more than one $x$-intercept. This occurs when the function has multiple values of $x$ at which the function has a value of zero.
Q: How do I determine if a function has an $x$-intercept?
A: To determine if a function has an $x$-intercept, you need to check if the function has a value of zero at any point. This can be done by plugging in values of $x$ into the function and checking if the function has a value of zero.
Q: What is the significance of the $x$-intercept in real-world applications?
A: The $x$-intercept is an important concept in many real-world applications, such as modeling population growth, modeling the spread of disease, and modeling the motion of objects.
Q: Can a function have a complex $x$-intercept?
A: Yes, a function can have a complex $x$-intercept. This occurs when the function has a value of zero at a complex number.
Q: How do I find the $x$-intercept of a function with a complex coefficient?
A: To find the $x$-intercept of a function with a complex coefficient, you need to use the quadratic formula and the properties of complex numbers.
Q: What is the relationship between the $x$-intercept and the roots of a polynomial?
A: The $x$-intercept of a polynomial is related to the roots of the polynomial. The roots of the polynomial are the values of $x$ at which the polynomial has a value of zero.
Q: Can a function have an $x$-intercept at infinity?
A: No, a function cannot have an $x$-intercept at infinity. The $x$-intercept is a point on the $x$-axis, and infinity is not a point on the $x$-axis.
Q: How do I determine if a function has an $x$-intercept at infinity?
A: To determine if a function has an $x$-intercept at infinity, you need to check if the function has a value of zero as $x$ approaches infinity.
Q: What is the significance of the $x$-intercept in calculus?
A: The $x$-intercept is an important concept in calculus, particularly in the study of limits and derivatives.
Q: Can a function have an $x$-intercept at a vertical asymptote?
A: No, a function cannot have an $x$-intercept at a vertical asymptote. The $x$-intercept is a point on the $x$-axis, and a vertical asymptote is a point on the $x$-axis where the function is undefined.
Q: How do I determine if a function has an $x$-intercept at a vertical asymptote?
A: To determine if a function has an $x$-intercept at a vertical asymptote, you need to check if the function has a value of zero at the vertical asymptote.
Q: What is the relationship between the $x$-intercept and the asymptotes of a function?
A: The $x$-intercept of a function is related to the asymptotes of the function. The asymptotes of the function are the lines that the function approaches as $x$ approaches infinity or negative infinity.
Q: Can a function have an $x$-intercept at a horizontal asymptote?
A: No, a function cannot have an $x$-intercept at a horizontal asymptote. The $x$-intercept is a point on the $x$-axis, and a horizontal asymptote is a line on the $y$-axis.
Q: How do I determine if a function has an $x$-intercept at a horizontal asymptote?
A: To determine if a function has an $x$-intercept at a horizontal asymptote, you need to check if the function has a value of zero at the horizontal asymptote.
Q: What is the significance of the $x$-intercept in statistics?
A: The $x$-intercept is an important concept in statistics, particularly in the study of regression analysis and time series analysis.
Q: Can a function have an $x$-intercept at a point of discontinuity?
A: No, a function cannot have an $x$-intercept at a point of discontinuity. The $x$-intercept is a point on the $x$-axis, and a point of discontinuity is a point where the function is undefined.
Q: How do I determine if a function has an $x$-intercept at a point of discontinuity?
A: To determine if a function has an $x$-intercept at a point of discontinuity, you need to check if the function has a value of zero at the point of discontinuity.
Q: What is the relationship between the $x$-intercept and the domain of a function?
A: The $x$-intercept of a function is related to the domain of the function. The domain of the function is the set of all possible values of $x$ for which the function is defined.
Q: Can a function have an $x$-intercept at a point outside the domain?
A: No, a function cannot have an $x$-intercept at a point outside the domain. The $x$-intercept is a point on the $x$-axis, and a point outside the domain is a point where the function is undefined.
Q: How do I determine if a function has an $x$-intercept at a point outside the domain?
A: To determine if a function has an $x$-intercept at a point outside the domain, you need to check if the function has a value of zero at the point outside the domain.
Q: What is the significance of the $x$-intercept in computer science?
A: The $x$-intercept is an important concept in computer science, particularly in the study of algorithms and data structures.
Q: Can a function have an $x$-intercept at a point of singularity?
A: No, a function cannot have an $x$-intercept at a point of singularity. The $x$-intercept is a point on the $x$-axis, and a point of singularity is a point where the function is undefined.
Q: How do I determine if a function has an $x$-intercept at a point of singularity?
A: To determine if a function has an $x$-intercept at a point of singularity, you need to check if the function has a value of zero at the point of singularity.
Q: What is the relationship between the $x$-intercept and the singularities of a function?
A: The $x$-intercept of a function is related to the singularities of the function. The singularities of the function are the points where the function is undefined.
Q: Can a function have an $x$-intercept at a point of oscillation?
A: No, a function cannot have an $x$-intercept at a point of oscillation. The $x$-intercept is a point on the $x$-axis, and a point of oscillation is a point where the function oscillates between two values.
Q: How do I determine if a function has an $x$-intercept at a point of oscillation?
A: To determine if a function has an $x$-intercept at a point of oscillation, you need to check if the function has a value of zero at the point of oscillation.
Q: What is the significance of the $x$-intercept in signal processing?
A: The $x$-intercept is an important concept in signal processing, particularly in the study of filters and transfer functions.
Q: Can a function have an $x$-intercept at a point of periodicity?
A: No, a function cannot have an $x$-intercept at a point of periodicity. The $x$-intercept is a point on the $x$-axis, and a point of periodicity is a point where the function repeats itself.