Problem 69 - Elementary and Intermediate Algebra

Question

Use a calculator to find each function value. Round to the nearest ten- thousandth. See Example 5 and Using Your Calculator \(g(x)=\sqrt[3]{x^{2}+1}\) a. \(g(6)\) b. \(g(21.57)\)

Step-by-Step Solution

Answer

\(g(6) \approx 3.3322\) and \(g(21.57) \approx 7.7584\).

Step 1: Understand the function

The function given is \(g(x) = \sqrt[3]{x^2 + 1}\). This means for any input \(x\), we will compute \(x^2 + 1\) and then take the cube root of that expression.

Step 2: Evaluate g(6)

First, calculate \(6^2 + 1\): \(6^2 = 36\), so \(6^2 + 1 = 37\). Next, find the cube root of 37 using a calculator. \(\sqrt[3]{37} \approx 3.3322\). Therefore, \(g(6) \approx 3.3322\).

Step 3: Evaluate g(21.57)

First, calculate \((21.57)^2 + 1\): \(21.57^2 = 465.8249\), so \(21.57^2 + 1 = 466.8249\). Next, find the cube root of this result using a calculator. \(\sqrt[3]{466.8249} \approx 7.7584\). Therefore, \(g(21.57) \approx 7.7584\).

Key Concepts

Understanding Cube Roots

When we talk about cube roots, we're looking for a number that, when multiplied by itself twice (i.e., cubed), gives the original number. In simpler terms, if you have a number, say 8, and take its cube root, you're asking: "What number, when multiplied by itself twice, equals 8?" The answer here would be 2, because \(2 \times 2 \times 2 = 8\).

In mathematical notation, the cube root is often denoted as \(\sqrt[3]{x}\). This symbol means you are finding the cube root of \(x\). For our exercise, you first calculate \(x^2 + 1\) and then find the cube root of that sum. Here's a quick recap of the steps:

• Calculate \(x^2 + 1\).
• Find the cube root of the result.

Understanding cube roots can be quite handy, especially when dealing with functions like \(g(x) = \sqrt[3]{x^2 + 1}\), where these steps become crucial for evaluating the output for any given \(x\).

Using a Calculator for Function Evaluation

Calculators are excellent tools for evaluating functions, especially when dealing with complex calculations like cube roots. When performing such operations, knowing how to use your calculator effectively makes a world of difference. Here’s how you can approach calculator usage for our function evaluation task:

• Input the expression inside the cube root. For example, \(x^2 + 1\) for the given \(x\).
• Use the calculator's cube root or \(\,^3\sqrt{}\) function to find the cube root of the result from the first step. Some calculators might require you to use a different key or menu option, so be sure you know how yours operates.
• Verify the calculator's settings to ensure it's configured to give answers to the desired decimal places—our task is rounding to the nearest ten-thousandth.

Using the calculator correctly streamlines solving equations like \(g(x)\) and feels rewarding by removing the guesswork and ensuring precise results.

Rounding Numbers Accurately

Rounding numbers is a crucial skill for making mathematics manageable and for representing results concisely. In general, rounding helps us approximate a number to a certain level of precision, which is especially handy when dealing with decimals that repeat or go on indefinitely. Here's a brief guide on rounding, particularly to the nearest ten-thousandth as required in our exercise:

• Identify the digit at the ten-thousandth place in your number. For example, in 3.33224, the 4 is in the ten-thousandth place.
• Look at the next digit to the right (in this case, the fifth decimal place). If it's 5 or greater, round up the digit in the ten-thousandth place. If it’s less than 5, keep the digit as it is.
• Apply this same rounding process for any number you encounter, ensuring results are consistently rounded according to your needs.

Rounding is key to maintaining a balance between precision and simplicity, especially in scientific calculations or reports where clarity and space are priorities.