A rational function is defined as the ratio of two polynomials, expressed in the form f(x) = P(x) / Q(x), where Q(x) is not zero. These functions are vital in calculus and algebra because they introduce concepts like discontinuities and asymptotic behavior.
Understanding Asymptotes
Asymptotes are lines that the graph of a function approaches but never touches as the input values reach certain limits. This calculator identifies two primary types:
Vertical Asymptotes: Occur where the denominator Q(x) equals zero (and the numerator is non-zero). They represent values where the function is undefined.
Horizontal Asymptotes: Determine the behavior of the function as x approaches infinity. They depend on the comparison between the degree of the numerator (n) and the degree of the denominator (m).
How to Determine Horizontal Asymptotes
Based on the degrees of the polynomials, we follow these rules:
If n < m: The horizontal asymptote is the x-axis, y = 0.
If n = m: The horizontal asymptote is the ratio of the leading coefficients, y = a/b.
If n = m + 1: There is no horizontal asymptote, but a slant (oblique) asymptote exists.
If n > m + 1: No horizontal or slant asymptote exists.
Example Calculation
Consider the function: f(x) = (x² – 4) / (x – 1)
Numerator: Coefficients [1, 0, -4] (Degree 2)
Denominator: Coefficients [1, -1] (Degree 1)
Y-Intercept: f(0) = (0-4)/(0-1) = 4.
Asymptote: Since degree of numerator (2) is exactly one more than denominator (1), it has a slant asymptote.
function calculateRationalFunction() {
var numInput = document.getElementById('numCoeff').value;
var denInput = document.getElementById('denCoeff').value;
// Parse coefficients
var numArr = numInput.split(',').map(function(item) { return parseFloat(item.trim()); }).filter(function(item) { return !isNaN(item); });
var denArr = denInput.split(',').map(function(item) { return parseFloat(item.trim()); }).filter(function(item) { return !isNaN(item); });
if (numArr.length === 0 || denArr.length === 0) {
alert("Please enter valid numeric coefficients separated by commas.");
return;
}
// Determine Degrees
var n = numArr.length – 1;
var m = denArr.length – 1;
// Calculate Y-Intercept (f(0) = a0 / b0)
var a0 = numArr[numArr.length – 1];
var b0 = denArr[denArr.length – 1];
var yIntStr = "";
if (b0 === 0) {
yIntStr = "None (Undefined at x=0)";
} else {
var val = a0 / b0;
yIntStr = "(0, " + val.toFixed(2) + ")";
}
// Determine Horizontal/Slant Asymptote
var hAsymptoteStr = "";
var leadingNum = numArr[0];
var leadingDen = denArr[0];
if (n < m) {
hAsymptoteStr = "y = 0";
} else if (n === m) {
var ratio = leadingNum / leadingDen;
hAsymptoteStr = "y = " + ratio.toFixed(2);
} else if (n === m + 1) {
hAsymptoteStr = "Slant Asymptote exists (Linear)";
} else {
hAsymptoteStr = "None";
}
// Simple Domain Logic (finding roots for linear/quadratic denominators)
var domainStr = "All real numbers except where the denominator is zero.";
// Display results
document.getElementById('rfResult').style.display = 'block';
document.getElementById('nDeg').innerText = n;
document.getElementById('mDeg').innerText = m;
document.getElementById('yIntercept').innerText = yIntStr;
document.getElementById('hAsymptote').innerText = hAsymptoteStr;
document.getElementById('domainRes').innerText = domainStr;
// Format function display
var fDisplay = "f(x) = (";
for(var i=0; i 0 ? "x" + (p > 1 ? "^"+p : "") : "");
if(i = 0) fDisplay += " + ";
else if (i < numArr.length – 1) fDisplay += " ";
}
fDisplay += ") / (";
for(var j=0; j 0 ? "x" + (q > 1 ? "^"+q : "") : "");
if(j = 0) fDisplay += " + ";
else if (j < denArr.length – 1) fDisplay += " ";
}
fDisplay += ")";
document.getElementById('functionDisplay').innerText = fDisplay;
}