How to Solve Rational Function Related Problems

A rational function is one in which the ratio of polynomials is the same. A rational function is one with only one variable, x, and may be written as f(x) = p(x) / q(x), where p(x) and q(x) are polynomials with q(x) is not equal to 0.

A rational function is the quotient of polynomials with a degree of at least one in the denominator. In other words, the denominator must contain a variable. When the denominator of a fraction equals 0, it is not defined. This is the crucial element in determining a rational function’s domain and range. 

The degree of a rational function has multiple non-equivalent definitions. When the fraction is reduced to the lowest terms, the degree of a rational function is usually the maximum of the degrees of its constituent polynomials P and Q. 

Solution of problems related to rational function:

• Problem 1: A big mixing tank now holds 100 gallons of water, which has been combined with 5 pounds of sugar. A tap will open, pouring 10 gallons of water per minute into the tank at the same time that 1 pound of sugar is put into the tank every minute. After 12 minutes, calculate the sugar concentration (pounds per gallon) in the tank. Is that a higher level of focus than at the start?

Solution: Let t be the time since the tap was turned on. These are constant rates of change since the water rises at 10 gallons per minute and the sugar increases at 1 pound per minute. This indicates that the amount of water in the tank, as well as the amount of sugar in the tank, is changing linearly. We may create an equation for each separately:

​water: S(t)=5+1t in pounds: W(t)=100+10t in gallons of sugar.

The concentration, C, will be the weight of sugar in pounds per gallon of water.

c = (5 + t) /(100 + 10t)

C(t) at t= 12 is used to calculate the concentration after 12 minutes.

C (12) = 17/ 220

This corresponds to 17 pounds of sugar every 220 gallons of water.

The concentration is initially high.

C (0) = 1/120

Because 17/220 ≈ 0.08 > 1/20 =0.05, the concentration is higher after 12 minutes than it was at the start.

• Problem 2: Sam can paint a house in less than five hours. Gary can finish it in four hours. How long will it take for the two to collaborate?

Solution: Sally works at a rate of S = Job done / 4

 John works at a rate of J = Job done/6 

Now the combined rate of doing the job is

S + J = Job/4 + Job/6 = 5 / 12 Job 

Now, Rate x Time = Job

Thus, 

Time = Job/ Rate 

This gives,

t = 12/5 = 2.4 hours = 2 hours + 24 minutes

hence, the total time required is 2 Hrs. and 24 minutes.

How to find a range of the rational Function?

Solution: A rational function’s range is the collection of all its outputs (y-values). To determine the range of a rational function y= f(x), use the following formula:

• Replace f(x) with y if it appears in the equation.
• For x, solve the equation.
• Set the resultant equation’s denominator to 0 and solve it for y.
• The range is the collection of all real numbers other than the y values given in the previous phase.

Conclusion:

• The form of a rational function equation is f(x) = P(x) / Q(x), where Q(x) 0.
• At least one vertical asymptote exists for every rational function.
• There is only one horizontal asymptote for any rational function.
• There is only one slant asymptote for every rational function.