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Manipulating Radical Functions

Method

Combining Radical Functions

Radical functions can be combined, through addition, subtraction and multiplication. Depending on the original functions and which operations are used, the resulting function can have similar or new characteristics.
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Exercise

Consider the functions. Find if

Show Solution
Solution
When functions containing one or more radicals are multiplied, it is often helpful to rewrite the expressions using rational exponents.
Next, the properties of exponents can be used to simplify the product.
Now that the function has been simplified completely, we can rewrite it with radical.
Concept

Inverse Functions

Inverse functions are two functions that undo each other. The functions and are inverses of each other if

When a function is denoted its inverse is often referred to as
Concept

Inverse Functions to Radical Functions

Radicals can be eliminated by raising them to the same power as the index of the radical. Therefore, the inverse function to a radical function is typically a power function. The radical function has the inverse function This can be proven by showing Consider I.

Rule

Next, consider II.

Rule

It has been shown that and are inverses of each other. This is true for all values of
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Exercise

Determine if the two functions are inverses of each other.

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Solution

The two functions, and are inverses if Thus, if we can show that

  1. , and
we can conclude that and are inverses. We'll begin with
Now that we've satisfied we'll attempt to satisfy
Since both and are inverses of each other.
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Exercise

The graph of the function is shown.

Use the graph to show that is invertible. Then, determine the inverse, and graph both functions.

Show Solution
Solution

Graphically, it can be shown that a function is invertible by performing the horizontal line test. If we move an imaginary horizontal line across the graph of a function, and the line intersects the graph more than once anywhere, the function is not invertible. Here, we'll draw more than one line to be sure.

The horizontal lines in the graph both intersect the function once. If we place a line anywhere else it still intersects the graph at the most once. Therefore, the function is invertible. Next, we can find the inverse, algebraically. To begin, replace with in the given rule. Next step is to switch and in the function rule. Now we need to solve for The resulting equation will be the inverse of the given function.
Thus, the inverse is We'll graph and in the same coordinate plane.

The domain of is restricted to As a consequence, the domain of must be restricted to the same interval, even though is defined for all values of

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