Problem 90 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\(g(1) = -\sqrt[3]{7}\)

Step 1: Identify the function for evaluation

We need to evaluate the function \(g(x) = \sqrt[3]{x-8}\) at \(x = 1\). This means replacing \(x\) in the function \(g(x)\) with 1.

Step 2: Substitute the value into the function

Substitute \(x = 1\) into the function \(g(x)\):\[g(1) = \sqrt[3]{1-8}\]

Step 3: Simplify the expression

Calculate the expression inside the cube root. \[1 - 8 = -7\]Now substitute back into the function to get:\[g(1) = \sqrt[3]{-7}\]

Step 4: Evaluate the cube root

The cube root of \(-7\) can be simplified as it is a real number, and the cube root of a negative number is also negative. Therefore,\[g(1) = -\sqrt[3]{7}\]

Key Concepts

Cube Roots

Cube roots are used to find a number that, when multiplied by itself three times, gives the original number. Unlike square roots, cube roots can handle negative numbers seamlessly. This is because multiplying three negative numbers results in a negative product. For instance, consider the number - **Example**: The cube root of -8 is -2, because \(-2 \times -2 \times -2 = -8\). Cube roots are denoted as \(\sqrt[3]{ }\). The expression \(\sqrt[3]{x}\) represents the cube root of \(x\). The cube root is helpful in handling equations and functions where the variable has an exponent of 3. Keep in mind that only cube roots of perfect cubes will yield an integer value, while others, like \(\sqrt[3]{7}\), will remain in their radical form unless approximated.

Radical Functions

Radical functions involve roots, such as square roots, cube roots, or higher-order roots. They take the form \(f(x) = \sqrt[n]{g(x)}\), where \(n\) defines the degree of the root, and \(g(x)\) is a function involving \(x\).These functions often appear in scenarios where a physical quantity depends on the root of another. Solving these can involve various algebraic techniques to simplify the expressions under the radical sign. For example, in the function \(g(x) = \sqrt[3]{x-8}\), as we evaluate at \(x=1\):

• First, substitute 1 for \(x\).
• Then, simplify the expression inside the cube root.
• Lastly, evaluate the cube root of the resulting number.

Radical functions can get complex but remain a critical part of algebra, allowing us to model diverse real-world phenomena.

Algebraic Expressions

Algebraic expressions are mathematical phrases that use numbers, variables, and operations to describe relationships. These expressions can be simple, such as \(x + 3\), or more complex, incorporating powers and roots, for example, \(\sqrt[3]{x-8}\). Understanding algebraic expressions is crucial for simplifying, evaluating, and solving equations or functions. When working with radical expressions, it's important to:

• Identify the parts of the expression: constants, variables, and operations
• Apply the correct operation order: exponents, roots, and other operations
• Look for opportunities to simplify radical parts when possible

Algebra serves as the foundation of higher mathematics, allowing us to express complex problems in simpler terms and perform function evaluations, just like finding the value of \(g(x)\) with a specific \(x\) value.