Problem 85 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( g(7) = -1 \).

Step 1: Identify the Function Formula for g(x)

The function given is \( g(x) = \sqrt[3]{x - 8} \). This means for any value of \( x \), you first subtract 8 from \( x \), then take the cube root of the result.

Step 2: Substitute the Value into the Function

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

Step 3: Simplify the Expression Inside the Cube Root

Calculate the expression inside the cube root: \( 7 - 8 = -1 \). Then the expression becomes \( \sqrt[3]{-1} \).

Step 4: Calculate the Cube Root

Determine the cube root of \(-1\). The cube root of \(-1\) is \(-1\) because \((-1)^3 = -1\). Thus, \( g(7) = -1 \).

Key Concepts

Function Evaluation

Function evaluation is a core concept in algebra that involves determining the value of a function for a specific input.
To effectively evaluate a function, follow these steps:

• Identify the Function: Begin by understanding the structure and formula of the function you are dealing with, such as linear, quadratic, or in our example, a cube root function.
• Substitute the Input Value: Replace the variable in the function with the number you want to evaluate. In this case, we needed to find the value of the function when the variable is 7.
• Simplify the Expression: Simplify any arithmetic operations to find the final result.
  This simplifies our way to evaluate the function and gets us to the correct answer.

Understanding how to evaluate functions is crucial as it allows us to predict how changing inputs will affect outputs.

Cube Roots

Cube roots may seem challenging, but they are quite an approachable topic. A cube root of a number is a value that, when multiplied by itself twice more, gives the original number.
In mathematical terms, for a number \[ a, \]\( \sqrt[3]{a} = b \) if \[ b^3 = a. \] To find the cube root of a number:

• Determine the Cube Root: Think of a number that, when multiplied by itself three times, results in the original number.
• Consider Negative Numbers: Don't forget that cube roots of negative numbers are also negative, like in our example, where \( \sqrt[3]{-1} = -1.\)

Cube roots let us solve equations involving cubes and explore volume problems in three dimensions.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra involving numbers, variables, and operations. Understanding them is essential to tackling function-based problems.
Here's what you need to know about working with algebraic expressions:

• Simplification: Combining like terms and applying arithmetic operations help simplify expressions, making them easier to work with.
• Substitution: Replacing variables with numbers, particularly when evaluating functions, is a common practice.
• Basic Arithmetic Operations: Learn addition, subtraction, multiplication, and division involving negative numbers and how to handle them in expressions.

Mastering algebraic expressions prepares you for more complex algebra topics and problem-solving in mathematics.