Problem 81 - Intermediate Algebra

Question

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(-19) $$

Step-by-Step Solution

Answer

The value of \( g(-19) \) is -3.

Step 1: Understand the Function

First, we need to understand what the function \( g(x) \) represents. The function is given by \( g(x) = \sqrt[3]{x - 8} \). This means that for any input \( x \), \( g(x) \) represents the cube root of \( x - 8 \).

Step 2: Substitute the Value

We need to find \( g(-19) \). This means we substitute \( x = -19 \) into the function \( g(x) = \sqrt[3]{x - 8} \). Hence, \( g(-19) = \sqrt[3]{-19 - 8} \).

Step 3: Simplify the Expression

Now, simplify the expression inside the cube root. \( -19 - 8 = -27 \). Therefore, we need to calculate the cube root of \(-27\): \[ g(-19) = \sqrt[3]{-27} \].

Step 4: Calculate the Cube Root

The cube root of \(-27\) is \(-3\) because \((-3) \times (-3) \times (-3) = -27\). Therefore, \( g(-19) = -3 \).

Key Concepts

Cube Root Function

Understanding the cube root function is crucial for solving problems like the one given in the exercise. A cube root function, denoted as \( \sqrt[3]{x} \), determines which number multiplied three times results in the given number, \( x \). This function can take any real number, positive or negative.
In the context of the function \( g(x) = \sqrt[3]{x - 8} \), it means that for any number you plug into \( x \), you first subtract 8, and then find the cube root of this result.

• Cube roots can involve both positive and negative numbers.
• A cube root undoes the effect of cubing a number.
• Unlike square roots, cube roots are defined for all real numbers, which allows negative inputs without resulting in imaginary numbers.

Substitution Method

The substitution method is a straightforward approach used in function evaluation where you replace a variable with a specific number. This method is crucial in evaluating functions at given points.
In our exercise, to find \( g(-19) \), the method involves plugging \(-19\) into \( g(x) = \sqrt[3]{x - 8} \).

• Locate the variable in the function.
• Replace this variable with the number you are evaluating the function at (e.g., \(-19\)).
• You will then perform any operations given in the function rule.

This method simplifies calculations and helps us focus on the arithmetic required for function evaluation.

Simplifying Expressions

Simplifying expressions is a necessary step to reduce complex mathematical expressions into more easily workable forms. This process involves breaking down the problem into simpler parts to evaluate or solve a function.
For the function \( g(-19) = \sqrt[3]{-19 - 8} \), start by simplifying the expression inside the cube root:
- Calculate \(-19 - 8\).

• This gives \(-27\), simplifying our task to finding the cube root of \(-27\).

Simplifying helps reduce errors and makes the next steps in solving a problem more intuitive.

Negative Numbers in Roots

Dealing with negative numbers in roots might initially seem tricky, but it is manageable with cube roots. Unlike square roots, cube roots can have negative results, and they are perfectly legitimate in real numbers.
In our example, \( \sqrt[3]{-27} = -3 \). Why? Because when you multiply \(-3 \times -3 \times -3\), you obtain \(-27\).

• Cube roots handle negative numbers because they involve an odd multiplication sequence. The final product can be negative since the factors multiplied are negative.
• This property makes cube roots versatile and applicable to a wider range of problems.

These characteristics make cube roots quite useful, especially for real-world applications that involve three-dimensional concepts or negative outputs.