Problem 84 - Intermediate Algebra

Question

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(1) $$

Step-by-Step Solution

Answer

\(g(1) = \sqrt[3]{-7}\), approximately \(-1.913\).

Step 1: Understand Function Definition

We are given the function \( g(x) = \sqrt[3]{x - 8} \). This means that for any input \( x \), the output is the cube root of the expression \( x - 8 \).

Step 2: Substitution of Value

To find \( g(1) \), we need to substitute \( 1 \) for \( x \) in the function. That is, compute \( g(1) = \sqrt[3]{1 - 8} \).

Step 3: Simplification of Expression

Calculate \( 1 - 8 \) to get \(-7\). Now, we have \( g(1) = \sqrt[3]{-7} \).

Step 4: Evaluating the Cube Root

The cube root of \(-7\) is simply \(-7\) raised to the power of \( \frac{1}{3} \), which is \( \sqrt[3]{-7} \). Since -7 is negative and cube roots of negative numbers are also negative, \( \sqrt[3]{-7} = -\sqrt[3]{7} = -1.913 \) approximately.

Key Concepts

Cube Root

A cube root is a mathematical operation that finds a number which, when multiplied by itself three times, gives the original number. Unlike square roots, cube roots can be calculated for negative numbers. Cube roots are often symbolized using the radical sign with a small three, like this: \( \sqrt[3]{x} \). In our context, finding \( g(1) = \sqrt[3]{1 - 8} \) required calculating the cube root of \(-7\).

• For instance, when you see \( \sqrt[3]{-7} \), you are finding the number that, when cubed, equals \(-7\). It's expressed as \( (-7)^{\frac{1}{3}} \).
• If \( x^3 = y \), then \( x = \sqrt[3]{y} \).

When dealing with cube roots, remember that they behave differently with negatives compared to square roots. This is because multiplying three negative numbers together results in a negative number. Consequently, \( \sqrt[3]{-7} \) remains negative and equates to approximately \(-1.913\). Understanding how cube roots work is essential for solving problems that involve cubic calculations.

Function Substitution

Function substitution is a method used to evaluate a function at a specific value of the variable. This involves replacing the variable with its given value and simplifying the resulting expression.

• Given a function \( g(x) = \sqrt[3]{x - 8} \), to find \( g(1) \), you substitute \( 1 \) in place of \( x \).
• This substitution gives you a new expression: \( \sqrt[3]{1 - 8} \).

The key is to carefully replace the variable inside the function while maintaining the structure of the function. This straightforward method is essential for evaluating function values, and forms a foundation for understanding more complex mathematical relationships.

Simplifying Expressions

Simplifying expressions means to perform all possible operations to make an expression as straightforward as possible. This involves combining like terms and simplifying numerical expressions. In our example, after substituting \( x = 1 \) into the function \( g(x) \), we have \( \sqrt[3]{1 - 8} \).

• Calculate \( 1 - 8 \), which simplifies the expression to \( \sqrt[3]{-7} \).
• This step reduces the expression to its core form where it can be evaluated or further manipulated.

Simplifying expressions is an invaluable skill in algebra as it allows you to reduce complex problems into simpler, more manageable forms. This is especially handy when dealing with both numerical and algebraic expressions. By honing this skill, functions, including those with cube roots, can be evaluated more efficiently.