How to Solve Irrational Functions?

Related Topics

• How to Approximate Irrational Numbers?

A step-by-step guide to irrational functions

An irrational function can be said to be a function that cannot be written as the quotient of two polynomials, but this definition is not usually used. In general, the most commonly used definition is that an irrational function is a function that contains variables in the radicals, i.e., square roots, cube roots, etc.

Therefore, the fundamental form of an irrational function is:

\(\color{blue}{f\left(x\right)=\sqrt[n]{\left(g\left(x\right)\right)^m}}\) or \(\color{blue}{f\left(x\right)=\left(g\left(x\right)\right)^{\left(\frac{m}{n}\right)}}\)

Where \(g(x)\) is a rational function.

• If the index \(n\) of the radical is odd, it is possible to calculate the domain of all real numbers.
• If the index \(n\) of the radical is even, we need \(g(x)\) to be positive or zero since the even roots of a negative number are not real.

Finding the Domain: Even vs. Odd Index

The index of the root decides the domain. Even roots of negative numbers are not real, but odd roots are.

The last row is the one students miss. An odd root accepts negative values, but the expression inside still has to make sense. Here you cannot divide by zero.

Solving an Equation with an Irrational Function

Solve √(x + 6) = x.

1. Isolate the radical. It is already alone on the left.
2. Square both sides. x + 6 = x².
3. Solve. x² − x − 6 = 0, so (x − 3)(x + 2) = 0, giving x = 3 or x = −2.
4. Check each answer in the original equation. For x = 3: √9 = 3, which works. For x = −2: √4 = 2, and 2 is not −2, so it fails.

The solution is x = 3. The value x = −2 is an extraneous solution. Squaring created it, because a square root can never equal a negative number. This is why the check step is not optional.

Frequently Asked Questions

What is an irrational function?

An irrational function is a function with the variable inside a radical, or raised to a fractional exponent. Examples include f(x) = √(x + 5), g(x) = ∛(4x) and h(x) = (x − 2)^(1/2). Many textbooks call these radical functions. A function like f(x) = √2 · x is not irrational, because the variable itself is not under the root.

What is the general form of an irrational function?

The general form is f(x) = ⁿ√g(x), where n is the index of the root (a whole number of 2 or more) and g(x) is an expression in x, often a polynomial. It can also be written as f(x) = g(x)^(1/n). For f(x) = ∛(x² − 1), the index is 3 and g(x) = x² − 1.

What is the domain of an irrational function?

It depends on the index. With an even index, like a square root, the expression under the radical must be 0 or more. For f(x) = √(6 − 2x), solve 6 − 2x ≥ 0 to get x ≤ 3. With an odd index, like a cube root, any real number works as long as g(x) is defined, so ∛(x − 9) has a domain of all real numbers.

How do you solve irrational functions?

Isolate the radical, raise both sides to the power of the index, solve, and then check every answer. For √(2x + 3) = x, squaring gives 2x + 3 = x², so x² − 2x − 3 = 0 and (x − 3)(x + 1) = 0. Checking shows x = 3 works, since √9 = 3, but x = −1 fails, since √1 = 1, not −1. The only solution is x = 3.

How do you solve radical functions graphically?

Graph each side of the equation as its own function and find where the graphs cross. The x-coordinate of each intersection point is a solution. To solve √(x + 1) = 2, graph y = √(x + 1) and y = 2; they meet at x = 3, and √4 = 2 confirms it. Graphing also shows right away when an equation has no real solution.

What does the graph of an irrational function look like?

A square root graph like y = √x starts at an endpoint, here (0, 0), and curves upward to the right, rising more slowly as x grows. Shifting the inside, as in √(x − 4), moves the starting point to (4, 0). A cube root graph like y = ∛x has no endpoint: it passes through the origin in a gentle S shape and covers all real x-values.