Problem 87 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the problem

We need to find the value of the function \( g(x) \) when \( x = -19 \). Given \( g(x) = \sqrt[3]{x-8} \), we need to substitute \( x = -19 \) into this expression.

Step 2: Substitute the value into the function

Substitute \( x = -19 \) into the expression for \( g(x) \): \[ g(-19) = \sqrt[3]{-19 - 8} \]

Step 3: Simplify inside the function

Subtract \( 8 \) from \( -19 \) inside the expression: \[ g(-19) = \sqrt[3]{-27} \]

Step 4: Calculate the cube root

Find the cube root of \( -27 \): The cube root of \( -27 \) is \( -3 \) (since \( (-3) \times (-3) \times (-3) = -27 \)). So, \[ g(-19) = -3 \]

Step 5: Final Answer

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

Key Concepts

Cube Root

The cube root of a number is an operation that reverses the effect of cubing a number. In simple terms, finding the cube root of a number is like asking, "What number, when multiplied by itself three times, gives me this original number?" For example, the cube root of 27 is 3, because when you multiply 3 by itself twice (3 \( \times \) 3 \( \times \) 3), you get 27.

Cube roots are important in various mathematical contexts, such as solving cubic equations, understanding geometric problems involving volume, or evaluating function values like in this exercise. When dealing with cube roots:

• They can be positive or negative. For instance, the cube root of -27 is -3 because \(-3 \times -3 \times -3 = -27\).
• If you encounter an expression like \( \sqrt[3]{x-a} \), you're looking at a cube root operation that involves a shift in the input value by \(a\).

In our problem, we calculated \( \sqrt[3]{-27} \) and concluded it simplified to -3, which directly comes from understanding how cube roots work.

Substitution

Substitution is a common algebraic method used to evaluate functions, expressions, or equations. It's like placing a value directly wherever specified by the variable within an expression. This is particularly handy when an equation involves a specific variable value you need to replace. Here's how you can think about substitution:

• Identify where in the equation you need to substitute the variable. In our function \( g(x) = \sqrt[3]{x - 8} \), we substitute \( x = -19 \).
• Replace every place you see that variable in the expression with the specific value. In this case, it involved writing \( g(-19) = \sqrt[3]{-19 - 8} \).
• Work through the algebra as usual to simplify and solve the expression after substitution has taken place.

Substitution is a straightforward yet powerful technique to solve real-world problems as it allows focusing on specific scenarios or data points by setting them directly in the equation.

Simplification

Simplification is the process of making a mathematical expression easier to understand and work with by reducing it to its simplest form. In the steps given, simplification includes performing basic arithmetic and combining like terms. Here's how it worked in the exercise:

• After completing substitution, we started with \( \sqrt[3]{-19 - 8} \). This is the first step where actual simplification occurred by evaluating \(-19 - 8\).
• We found \(-19 - 8\) equals \(-27\), so \( g(-19) = \sqrt[3]{-27} \).
• Finding \( \sqrt[3]{-27} \) was our last simplification step, arriving at the final answer \(-3\).

Simplification is crucial because it allows us to work with expressions in their most manageable form, making subsequent calculations or interpretations much easier. It is especially useful for reducing complex equations or expressions to a form where insights and results can be easily drawn.