Category: Calculus
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Asymptotes of Rational Function
A rational function can have three types of asymptotes: horizontal, vertical, and slant asymptotes. Apart from these, it can have holes as well. Let us see how to find each of them. Holes of a Rational Function The holes of a rational function are points that seem that they are present on the graph of the…
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Range of Rational Function
The range of a rational function is the set of all outputs (y-values) that it produces. To find the range of a rational function y= f(x): If we have f(x) in the equation, replace it with y. Solve the equation for x. Set the denominator of the resultant equation ≠ 0 and solve it for…
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Domain and Range of Rational Function
Any fraction is not defined when its denominator is equal to 0. This is the key point that is used in finding the domain and range of a rational function. Domain of Rational Function The domain of a rational function is the set of all x-values that the function can take. To find the domain of a rational function…
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How to Identify a Rational Function?
By the definition of the rational function (from the previous section), if either the numerator or denominator is not a polynomial, then the fraction formed does NOT represent a rational function. For example, f(x) = (4 + √x)/(2-x), g(x) = (3 + (1/x)) / (2 – x), etc are NOT rational functions as numerators in…
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What is a Rational Function?
A rational function is a function that is the ratio of polynomials. Any function of one variable, x, is called a rational function if, it can be represented as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials such that q(x) ≠ 0. For example, f(x) = (x2 + x – 2) / (2×2 – 2x – 3) is a rational…
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Rational Function
A rational function is a ratio of polynomials where the polynomial in the denominator shouldn’t be equal to zero. Isn’t it resembling the definition of a rational number (which is of the form p/q, where q ≠ 0)? Did you know Rational functions find application in different fields in our day-to-day life? Not only do…
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Algebraic Identities of Complex Numbers
All the algebraic identities apply equally for complex numbers. The addition and subtraction of complex numbers and with exponents of 2 or 3 can be easily solved using algebraic identities of complex numbers. (z1+z2)2=z21+2z1z2+z22(z1+z2)2=z12+2z1z2+z22 (z1−z2)2=z21−2z1z2+z22(z1−z2)2=z12−2z1z2+z22 (z1+z2)3=z31+3z21z2+3z1z22+z32(z1+z2)3=z13+3z12z2+3z1z22+z23 (z1−z2)3=z31−3z21z2+3z1z22−z32(z1−z2)3=z13−3z12z2+3z1z22−z23 (z1+z2)(z1−z2)=z21−z22(z1+z2)(z1−z2)=z12−z22 (z1+z2+z3)2=z21+z22+z23+2z1z2+2z2z3+2z3z1
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Operations on Complex Numbers
The various operations of addition, subtraction, multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows. Addition of Complex Numbers Th addition of complex numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added…
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Ordering of Complex Numbers
The ordering of complex numbers is not possible. Real numbers and other related number systems can be ordered, but complex numbers cannot be ordered. The complex numbers do not have the structure of an ordered field, and there is no ordering of the complex numbers that are compatible with addition and multiplication. Also, the non-trivial…
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Equality of Complex Numbers
The equality of complex numbers is similar to the equality of real numbers. Two complex numbers z1=a1+ib1z1=a1+ib1 and z2=a2+ib2z2=a2+ib2 are said to be equal if the rel part of both the complex numbers are equal a1=a2a1=a2, and the imaginary parts of both the complex numbers are equal b1=b2b1=b2. Also, the two complex numbers in the polar form are equal, if and…