7.5: Further Exploration with Radicals

section 7.5 Learning Objectives

7.5: Further Exploration with Radicals

  • Use the Pythagorean Theorem to solve applications involving right triangles
  • Find the domain of a radical function

This section will discuss applications which use square roots, in particular the Pythagorean Theorem. As always, the following steps will help to translate and solve the problem.

1. Read through the entire problem

2. Organize the information (a picture may be useful)

3. Write the equation

4. Solve the equation

5. Check the answer

Pythagorean Theorem

If we are given the measurements of two sides of a right triangle (A right triangle is one where one of the angles is 90 degrees; indicated with a square in the angle), we can find the measurement of the third side by using the Pythagorean Theorem. The Pythagorean Theorem states that the square of hypotenuse is equal to the sum of the squares of the other two sides.

The Pythagorean Theorem formula is:

[latex]a^2+b^2=c^2[/latex]   where c is the hypotenuse; a and b are the legs. 

Right triangle labeled

The video below will walk you through the proof and a few examples of using The Pythagorean Theorem, including an application example.

Example 1

Find the length of the hypotenuse of the right triangle pictured below:

right triangle with legs 7 and 24

Example 2

Find the missing length of the leg of the right triangle pictured below:

right triangle with hypotenuse=13 and leg=5

The Pythagorean Theorem can be used in a variety of real-world applications. The next two examples show some simple applications.

Example 3

Given a rectangular picture frame whose opening is 12 inches by 10 inches, how long is the diagonal from one corner of the opening to the other? Give both an exact answer and an approximation to the nearest hundredth. 

Example 4

A 6foot ladder is placed against a wall. If the base of the ladder is 3 feet from the wall, how high up the wall will the ladder reach? Give both an exact answer and an approximation to the nearest hundredth. 

The next example includes a radical in one of the lengths. See if you can apply the skills of this chapter to this problem.

Example 5

A right triangle’s hypotenuse is 8 meters, and one leg is [latex]4\sqrt{3}[/latex] meters.  Find the length of the other leg.  

As mentioned earlier, the Pythagorean Theorem is used in many applications. In fact, it has played a critical role in real-world applications for centuries! If you are interested in exploring another real-world example, celestial navigation and the mathematics behind it, watch this video for fun.

Domain of Radical Functions

In module 3, we introduced the domain of a function as the set of input values that produce valid outputs.  In this module, we have been exploring radical expressions.  In the next example, let us examine a radical function and determine if any domain issues arise.

Example 6

Consider the radical function [latex]f(x)=\sqrt{3x-2}[/latex]

A.  Compute [latex]f(1)[/latex], [latex]f(6)[/latex], [latex]f(7)[/latex], [latex]f\left(\frac{2}{3}\right)[/latex], and [latex]f(0)[/latex]. Round to three decimal places if needed.

B.  Determine if each of the [latex]x[/latex]-values from part A is in the domain of the function.

The previous example shows us that any input that results in a negative radicand is excluded from the domain. In other words, we must always ensure that the radicand is nonnegative ([latex]\geq 0[/latex]).

Finding the domain of a radical function with an even index

  • Set the radicand (the expression under the radical) to be [latex]\geq 0[/latex].
  • Solve the corresponding inequality for the variable.
  • Write your answer in the appropriate notation.
So now, let us find the domain of the radical function given in the previous example.

Example 7

Find the domain of the radical function [latex]f(x)=\sqrt{3x-2}[/latex]. Give your answer in set-builder notation and interval notation.

It is worth noting that had the function in the previous example been a fourth root, sixth root, or any even root, the process and final answer would not have changed. However, we saw in the beginning of this module that odd roots do not have the same issue with negative radicands.  For example, [latex]\sqrt[3]{-8}=-2[/latex]. It follows that odd roots will not have any domain issues.

Domain of a radical function with an odd index

  • The domain will be all real numbers. In interval notation, this is [latex](-\infty,\infty)[/latex].

Let us conclude the section with one more example to emphasize the difference between these two cases.

Example 8

Find the domain in set-builder notation for each radical function.  If the domain is all real numbers, state this.

A.  [latex]f(x)=\sqrt{5-x}[/latex]

B.  [latex]g(x)=\sqrt[3]{5-x}[/latex]