Here are the steps for graphing a rational function: Identify and draw the vertical asymptote using a dotted line. Identify and draw the horizontal asymptote using a dotted line. Plot the holes (if any) Find x-intercept (by using y = 0) and y-intercept (by x = 0) and plot them. Draw a table of two columns x and y… Continue reading Graphing Rational Functions
Horizontal Asymptote of a Rational Function
A horizontal asymptote (HA) of a function is an imaginary horizontal line to which its graph appears to be very close but never touch. It is of the form y = some number. Here, “some number” is closely connected to the excluded values from the range. A rational function can have at most one horizontal asymptote. Easy… Continue reading Horizontal Asymptote of a Rational Function
Vertical Asymptote of a Rational Function
A vertical asymptote (VA) of a function is an imaginary vertical line to which its graph appears to be very close but never touch. It is of the form x = some number. Here, “some number” is closely connected to the excluded values from the domain. But note that there cannot be a vertical asymptote at x… Continue reading Vertical Asymptote of a Rational Function
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… Continue reading Asymptotes of Rational Function
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… Continue reading Range of Rational Function
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… Continue reading Domain and Range of Rational Function
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… Continue reading How to Identify a Rational Function?
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… Continue reading What is a Rational Function?
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… Continue reading Rational Function
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