Refresher - Radical Functions

A radical function is a function defined by a radical expression. ${\displaystyle f\left(x\right)=\sqrt{x}}$is a square root function.

To find the domain of a radical function, we set whatever is inside the radical sign equal to or greater than zero, and then solve. Since ${\displaystyle f\left(x\right)=\sqrt{x}}$, then x≥0. So the domain will be all positive real numbers, [0. $\infty$). What this tells us is that when selecting points to graph this function, only non-negative values for x can be used because we cannot take the square root of a negative number. Since ${\displaystyle \sqrt{x}}$ is the principal (positive) square root of x, it always has a nonnegative value and therefore the range is also [0, $\infty$).

You may have noticed that the square root function has the shape of a portion of a rotated parabola. This is because the square root function, ${\displaystyle f\left(x\right)=\sqrt{x}}$, with its restricted domain, ${\displaystyle x\ge 0}$, is the inverse function of the restricted function, ${\displaystyle f\left(x\right)=x^2}$, where ${\displaystyle x\ge 0}$. Remember, the function ${\displaystyle f\left(x\right)=x^2}$ is not a one-to-one function, therefore does not have an inverse. However, by restricting the domain of ${\displaystyle f\left(x\right)=x^2}$ to only the positive values of ${\displaystyle x}$, we can see in the graph below that the inverse is ${\displaystyle f\left(x\right)=\sqrt{x}}$.

A radical function could also be a cube root function such as ${\displaystyle f\left(x\right)=\sqrt[3]{x}}$. To find the domain of this function, we consider what the values of x could be. Since a real number, positive, negative, or 0 can be used in the cube root function, ${\displaystyle f\left(x\right)=\sqrt[3]{x}}$ could be positive, negative, or 0. So, both the domain and the range of the cube root function are ${\displaystyle (-\infty ,\infty )}$.

Math 1A/1B: Pre-Calculus by Dr. Sarah Eichorn and Dr. Rachel Lehman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Links to an external site..

Solving Radical Equations

A radical equation is an equation in which a variable appears under a radical sign, such as $\sqrt{3x+1}=5$. Note that $x+\sqrt{3}=10$ is not a radical equation.

When working with radical equations, beware of extraneous solutions! Consider the equation, x = 2. There is only one solution {2}. If we square both sides of this equation, we get ${\displaystyle x^2=4}$. This new squared equation has two solutions, {-2, 2}. Note that the solution to the original equation, 2, is also a solution of the new squared equation. However, the new squared equation has a solution, -2, which is not a solution of the original equation. What this means is that not all solutions of the new equation are necessarily solutions of the original equation.

Solutions that do not satisfy the original equation are called extraneous solutions and must be discarded. How do we find extraneous solutions? We must check that all solutions of the new equation satisfy the original equation.

Math 1A/1B: Pre-Calculus by Dr. Sarah Eichorn and Dr. Rachel Lehman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Links to an external site..