Determine The Zeros Of The Function:12. $y=\frac{(x+3)(x-2)}{(x-1)(x+5)}$Zeros: $x=$
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Understanding Rational Functions
A rational function is a type of function that is defined as the ratio of two polynomials. In other words, it is a function that can be expressed as the quotient of two algebraic expressions. Rational functions are commonly used in mathematics and have numerous applications in various fields, including physics, engineering, and economics.
The Given Function
The given rational function is:
This function has two linear factors in the numerator and two linear factors in the denominator. To determine the zeros of this function, we need to find the values of x that make the numerator equal to zero.
Finding the Zeros of the Numerator
The numerator of the function is . To find the zeros of the numerator, we need to set this expression equal to zero and solve for x.
Using the zero-product property, we can set each factor equal to zero and solve for x.
Solving for x, we get:
These are the zeros of the numerator.
Finding the Zeros of the Denominator
The denominator of the function is . To find the zeros of the denominator, we need to set this expression equal to zero and solve for x.
Using the zero-product property, we can set each factor equal to zero and solve for x.
Solving for x, we get:
These are the zeros of the denominator.
Determining the Zeros of the Function
Now that we have found the zeros of the numerator and the denominator, we can determine the zeros of the function. The zeros of the function are the values of x that make the numerator equal to zero, but not the values of x that make the denominator equal to zero.
In other words, the zeros of the function are the values of x that satisfy the following condition:
Using the zeros of the numerator, we can write:
But we need to exclude the values of x that make the denominator equal to zero. Therefore, we exclude x=1 and x=-5.
The final answer is:
These are the zeros of the function.
Conclusion
In this article, we have determined the zeros of the given rational function. We have found the zeros of the numerator and the denominator, and then used these values to determine the zeros of the function. The zeros of the function are the values of x that make the numerator equal to zero, but not the values of x that make the denominator equal to zero.
References
- [1] "Rational Functions." MathWorld, Wolfram Research.
- [2] "Zeros of a Rational Function." Math Open Reference, 2023.
Further Reading
- "Rational Functions: A Tutorial." Math Is Fun, 2023.
- "Zeros of a Rational Function: A Guide." Purplemath, 2023.
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Q: What is a rational function?
A: A rational function is a type of function that is defined as the ratio of two polynomials. In other words, it is a function that can be expressed as the quotient of two algebraic expressions.
Q: How do I determine the zeros of a rational function?
A: To determine the zeros of a rational function, you need to find the values of x that make the numerator equal to zero, but not the values of x that make the denominator equal to zero.
Q: What is the difference between the zeros of the numerator and the denominator?
A: The zeros of the numerator are the values of x that make the numerator equal to zero. The zeros of the denominator are the values of x that make the denominator equal to zero. The zeros of the function are the values of x that make the numerator equal to zero, but not the values of x that make the denominator equal to zero.
Q: How do I find the zeros of the numerator?
A: To find the zeros of the numerator, you need to set the numerator equal to zero and solve for x. You can use the zero-product property to set each factor equal to zero and solve for x.
Q: How do I find the zeros of the denominator?
A: To find the zeros of the denominator, you need to set the denominator equal to zero and solve for x. You can use the zero-product property to set each factor equal to zero and solve for x.
Q: What if the numerator and denominator have common factors?
A: If the numerator and denominator have common factors, you need to cancel out these common factors before determining the zeros of the function.
Q: Can a rational function have no zeros?
A: Yes, a rational function can have no zeros. This occurs when the numerator is a constant and the denominator is a non-zero polynomial.
Q: Can a rational function have an infinite number of zeros?
A: No, a rational function cannot have an infinite number of zeros. This is because the numerator and denominator are both polynomials, and polynomials have a finite number of zeros.
Q: How do I graph a rational function?
A: To graph a rational function, you need to find the zeros of the numerator and the denominator, and then use these values to determine the x-intercepts and vertical asymptotes of the graph.
Q: What is the difference between a rational function and a polynomial function?
A: A rational function is a function that is defined as the ratio of two polynomials, while a polynomial function is a function that is defined as the sum of a finite number of terms, each of which is a constant times a power of x.
Q: Can a rational function be used to model real-world phenomena?
A: Yes, rational functions can be used to model real-world phenomena, such as the motion of an object under the influence of gravity, the growth of a population, and the behavior of a circuit.
Q: What are some common applications of rational functions?
A: Some common applications of rational functions include:
- Modeling the motion of an object under the influence of gravity
- Modeling the growth of a population
- Modeling the behavior of a circuit
- Solving problems in physics, engineering, and economics
Q: How do I determine the domain of a rational function?
A: To determine the domain of a rational function, you need to find the values of x that make the denominator equal to zero, and then exclude these values from the domain.
Q: What is the difference between the domain and the range of a rational function?
A: The domain of a rational function is the set of all possible input values, while the range is the set of all possible output values.
Q: Can a rational function have a domain that is not a subset of the real numbers?
A: No, a rational function cannot have a domain that is not a subset of the real numbers. This is because the input values of a rational function are always real numbers.
Q: How do I determine the range of a rational function?
A: To determine the range of a rational function, you need to find the values of y that correspond to each value of x in the domain.
Q: What is the difference between the range and the codomain of a rational function?
A: The range of a rational function is the set of all possible output values, while the codomain is the set of all possible output values, including those that are not actually attained by the function.
Q: Can a rational function have a range that is not a subset of the real numbers?
A: No, a rational function cannot have a range that is not a subset of the real numbers. This is because the output values of a rational function are always real numbers.
Q: How do I graph the range of a rational function?
A: To graph the range of a rational function, you need to find the values of y that correspond to each value of x in the domain, and then plot these points on a coordinate plane.
Q: What is the difference between the graph of a rational function and the graph of its range?
A: The graph of a rational function is a visual representation of the function, while the graph of its range is a visual representation of the set of all possible output values.
Q: Can a rational function have a graph that is not a function?
A: No, a rational function cannot have a graph that is not a function. This is because a rational function is a function, by definition.
Q: How do I determine the inverse of a rational function?
A: To determine the inverse of a rational function, you need to swap the x and y variables, and then solve for y.
Q: What is the difference between the inverse of a rational function and the function itself?
A: The inverse of a rational function is a function that undoes the action of the original function, while the function itself is a function that takes input values and produces output values.
Q: Can a rational function have an inverse that is not a rational function?
A: No, a rational function cannot have an inverse that is not a rational function. This is because the inverse of a rational function is always a rational function.
Q: How do I determine the composition of two rational functions?
A: To determine the composition of two rational functions, you need to substitute the output of the first function into the input of the second function.
Q: What is the difference between the composition of two rational functions and the product of two rational functions?
A: The composition of two rational functions is a function that takes input values and produces output values, while the product of two rational functions is a function that takes input values and produces output values.
Q: Can a rational function have a composition that is not a rational function?
A: No, a rational function cannot have a composition that is not a rational function. This is because the composition of two rational functions is always a rational function.
Q: How do I determine the derivative of a rational function?
A: To determine the derivative of a rational function, you need to use the power rule and the quotient rule.
Q: What is the difference between the derivative of a rational function and the function itself?
A: The derivative of a rational function is a function that represents the rate of change of the original function, while the function itself is a function that takes input values and produces output values.
Q: Can a rational function have a derivative that is not a rational function?
A: No, a rational function cannot have a derivative that is not a rational function. This is because the derivative of a rational function is always a rational function.
Q: How do I determine the integral of a rational function?
A: To determine the integral of a rational function, you need to use the power rule and the substitution method.
Q: What is the difference between the integral of a rational function and the function itself?
A: The integral of a rational function is a function that represents the accumulation of the original function, while the function itself is a function that takes input values and produces output values.
Q: Can a rational function have an integral that is not a rational function?
A: No, a rational function cannot have an integral that is not a rational function. This is because the integral of a rational function is always a rational function.
Q: How do I determine the area under a rational function?
A: To determine the area under a rational function, you need to use the definite integral.
Q: What is the difference between the area under a rational function and the function itself?
A: The area under a rational function is a measure of the accumulation of the function, while the function itself is a function that takes input values and produces output values.
Q: Can a rational function have an area under it that is not a rational function?
A: No, a rational function cannot have an area under it that is not a rational function. This is because the area under a rational function is always a rational function.