If the degree of the polynomial is odd, the range is all real numbers - for example, y = x5.
If the degree is even, use derivatives to find maxima or minima. You learn about derivatives, maxima and minima in any basic calculus course. Example:
y = x4 - 3x3
Take the derivative:
y' = 4x3 - 9x2
Solve for zero:
4x3 - 9x2 = 0
This will give you two maxima or minima; in this case, check at which of these points the function has the smallest value. Because of the positive coefficient of the leading term, the function values go from this point all the way to plus infinity.
The domain of a rational function is the whole of the real numbers except those points where the denominator of the rational function, simplified if possible, is zero.
To find the range of a rational function, one must analyze the behavior of the function as the input values approach different limits, particularly the vertical and horizontal asymptotes. It's crucial to identify any values that the function cannot output, which may occur due to restrictions from the denominator. Additionally, graphing the function can provide visual insights into the range, revealing intervals of output values. Ultimately, the range is determined by the values the function can take, considering any asymptotic behavior and discontinuities.
Yes. Rational functions must contain rational expressions in order to be rational.
y = cuberoot(x) for real x is not a rational function.
A rational function is the quotient of two polynomial functions.
A number does not have a range and domain, a function does.
The domain of a rational function is the whole of the real numbers except those points where the denominator of the rational function, simplified if possible, is zero.
To find the range of a rational function, one must analyze the behavior of the function as the input values approach different limits, particularly the vertical and horizontal asymptotes. It's crucial to identify any values that the function cannot output, which may occur due to restrictions from the denominator. Additionally, graphing the function can provide visual insights into the range, revealing intervals of output values. Ultimately, the range is determined by the values the function can take, considering any asymptotic behavior and discontinuities.
Yes. Rational functions must contain rational expressions in order to be rational.
Yes. Rational functions must contain rational expressions in order to be rational.
Yes. Rational functions must contain rational expressions in order to be rational.
Yes. Rational functions must contain rational expressions in order to be rational.
True
a rational expression.
y = cuberoot(x) for real x is not a rational function.
It is true that a rational function is a function whose equation contains a rational expression. This is used in various math classes.
The domain of the function 1/2x is {0, 2, 4}. What is the range of the function?