Problem 79 - Intermediate Algebra

Question

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(7) $$

Step-by-Step Solution

Answer

g(7) = -1

Step 1: Identify the function for g(x)

The function given for \( g(x) \) is \( g(x) = \sqrt[3]{x-8} \). This function represents the cube root of \( x \) minus 8.

Step 2: Substitute x in the function

Substitute \( x = 7 \) into the function \( g(x) = \sqrt[3]{x-8} \). This gives us the expression \( g(7) = \sqrt[3]{7-8} \).

Step 3: Simplify the expression

Simplify the expression \( \sqrt[3]{7-8} \). We have \( 7 - 8 = -1 \), so the expression becomes \( \sqrt[3]{-1} \).

Step 4: Evaluate the cube root

Evaluate \( \sqrt[3]{-1} \). The cube root of \(-1\) is \(-1\), because \((-1) \times (-1) \times (-1) = -1\). Thus, \( g(7) = -1 \).

Key Concepts

Cube Root Function

The cube root function, denoted as \( g(x) = \sqrt[3]{x-8} \), is an integral part of solving the original problem. Unlike square root functions, which provide only the non-negative root, cube root functions always provide a real number result, even for negative values. This is because any real number, whether positive or negative, multiplied by itself three times can have a real root. Thus, for the given function \( g(x) = \sqrt[3]{x-8} \), the expression inside the cube root, \( x - 8 \), determines the evaluation for any input \( x \). Understanding cube root functions is vital as they appear in solving a variety of mathematical problems, especially those involving symmetric equations and geometrical solutions. They also provide insights into transformations and translations of graphs, as the expression inside adjusts the entire curve either upwards, downwards, or sideways.

Substitution Method

Substitution is a fundamental technique in algebra that simplifies complex expressions. In the problem, to evaluate \( g(7) \), we substitute \( 7 \) for \( x \) in the function \( g(x) = \sqrt[3]{x-8} \). This substitution leads to \( g(7) = \sqrt[3]{7-8} \). By replacing \( x \) with a specific numerical value, we convert a general expression into a specific one which is easier to analyze and solve.
Substitution is essential for solving equations, evaluating definite integrals in calculus, and algorithm design in computer science. This method effectively breaks down complex problems, reducing them to simpler parts that are easier to manage. When dealing with functions, especially those with roots, substitution helps in isolating terms to clarify what operation must be performed next.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. In this exercise, once \( x = 7 \) is substituted into \( g(x) \), the expression becomes \( \sqrt[3]{7-8} \). Here, simplification occurs by calculating \( 7 - 8 \), which equals \(-1\). So, the expression now simplifies to \( \sqrt[3]{-1} \). Simplifying algebraic expressions is crucial in mathematics as it makes calculations more manageable and solutions clearer.
Some basic rules that assist with algebraic simplification include combining like terms, using the distributive property to eliminate parentheses, and reducing fractions to their simplest form. It is often the final step in problem-solving, leaving you with an understandable outcome, whether that's a numerical result or a simplified algebraic expression.

Negative Numbers in Functions

Handling negative numbers is a crucial skill when evaluating functions, particularly for roots. In this problem, you encounter \( \sqrt[3]{-1} \), the cube root of \(-1\). Unlike square roots, cube roots of negative numbers exist because multiplying a negative number an odd number of times results in a negative product. Therefore, \( \sqrt[3]{-1} \) simplifies to \(-1\) since \((-1) \times (-1) \times (-1) = -1\).
Dealing with negative numbers accurately is important not just in algebra, but also in calculus, statistics, and real-world problem-solving. Remember that in arithmetic, negative signs indicate direction or a lack of something, so it is crucial to carefully interpret them in problems.

• They can change the inequality or direction of inequalities.
• They affect calculations involving absolute values significantly.

Mastery of negative numbers in function evaluation leads to more confident handling of complex mathematical expressions.