Problem 65 - Elementary and Intermediate Algebra

Question

Find each function value, if possible. Do not use a calculator. See Example 5 . \(g(x)=\sqrt[3]{x-4}\) a. \(g(12)\) b. \(g(-23)\)

Step-by-Step Solution

Answer

a. \( g(12) = 2 \), b. \( g(-23) = -3 \).

Step 1: Substitute in Function for g(12)

Start by substituting 12 into the function for \( g(x) \). We have:\[ g(12) = \sqrt[3]{12 - 4} \]. Simplify inside the cube root: \(12 - 4 = 8\).

Step 2: Evaluate Cube Root for g(12)

Now find the cube root of 8. Since \( 2^3 = 8 \), we have:\[ g(12) = 2 \]. Thus, \( g(12) \) is 2.

Step 3: Substitute in Function for g(-23)

Next, substitute -23 into the function for \( g(x) \). We have:\[ g(-23) = \sqrt[3]{-23 - 4} \]. Simplify inside the cube root: \(-23 - 4 = -27\).

Step 4: Evaluate Cube Root for g(-23)

Now find the cube root of -27. Since \( (-3)^3 = -27 \), we have:\[ g(-23) = -3 \]. Thus, \( g(-23) \) is -3.

Key Concepts

Cube Root

A cube root is a special value that, when multiplied by itself three times, gives the original number. In mathematical terms, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \) and is the value \( y \) such that \( y^3 = x \). For example, if we consider \( 8 \), the cube root of \( 8 \) is \( 2 \) because \( 2 \times 2 \times 2 = 8 \).
For negative numbers, cube roots act similarly. The cube root of \( -27 \) is \( -3 \), since \( (-3) \times (-3) \times (-3) = -27 \). Unlike square roots, cube roots can be negative without issue, because a negative number raised to an odd power will remain negative.
This is important to grasp because understanding cube roots helps perform tasks such as simplifying expressions within functions like our example \( g(x)=\sqrt[3]{x-4} \), where knowing this operation allows us to solve for function values without a calculator.

Substitution in Functions

Substitution is a strategy to evaluate functions by replacing the variable with a specific value. By substituting, we are essentially plugging a number into the function and calculating the result. Consider the function \( g(x) = \sqrt[3]{x - 4} \).
Suppose we want to evaluate it at \( x = 12 \). We substitute 12 into the variable \( x \) which gives us \( g(12) = \sqrt[3]{12 - 4} \). This simplifies to \( \sqrt[3]{8} \), easily evaluated using cube roots.
Substitution is particularly useful in function evaluation because it transforms abstract expressions into concrete numbers, making the problem more manageable. Remember to always simplify the expression inside the function before calculating the result, as shown in our example.

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to work with by combining like terms and performing any basic arithmetic. In the context of functions like \( g(x)=\sqrt[3]{x-4} \), simplifying occurs right after the substitution step.
For instance, when substituting \( x = 12 \), we first compute \( 12 - 4 \), which simplifies the expression inside the cube root to 8. This makes subsequent calculations straightforward, like finding the cube root of the simplified result.
The same process is applied to negative numbers: substituting \( x = -23 \) gives \( -23 - 4 = -27 \). By handling simplification first, we can more easily determine the cube root of negative numbers as well. Regularly practicing simplification helps ensure that errors are minimized during more complex calculations.