Problem 7 - Calculus with Concepts in Calculus

Question

Find the numerical value of the function at the given values of \(a\). $$ g(t)=\sqrt[3]{t} ; a=27,-\frac{1}{8} $$

Step-by-Step Solution

Answer

\( g(27) = 3 \) and \( g(-\frac{1}{8}) = -\frac{1}{2} \).

Step 1: Understand the Function

The function given is \( g(t) = \sqrt[3]{t} \), which means we are finding the cube root of \( t \). We are asked to evaluate this function at different values of \( t \).

Step 2: Evaluate the Function at \( t = 27 \)

For the first value, \( t = 27 \), substitute \( t \) in the function: \( g(27) = \sqrt[3]{27} \). The cube root of 27 is 3 because \( 3^3 = 27 \). Thus, \( g(27) = 3 \).

Step 3: Evaluate the Function at \( t = -\frac{1}{8} \)

Next, substitute \( t = -\frac{1}{8} \) into the function: \( g\left(-\frac{1}{8}\right) = \sqrt[3]{-\frac{1}{8}} \). The cube root of \(-\frac{1}{8}\) is \(-\frac{1}{2}\) because \((-\frac{1}{2})^3 = -\frac{1}{8}\). So, \( g\left(-\frac{1}{8}\right) = -\frac{1}{2} \).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a particular input. To evaluate a function, you need to substitute the given value into the function's equation. This helps in determining how the value affects the overall result.
Here's a simple way to think about it:

• Identify the function and the input value you need to plug in.
• Replace the variable in the function with your given value.
• Perform any necessary calculations to find the function's output.

In our exercise, the task was to evaluate the cube root function at specific values. By substituting these values into the expression for the function, we determined the numerical outputs easily.

Cube Root Function

The cube root function is a type of radical function, where the output is derived from taking the cube root of a given value. Understanding how the cube root function works is essential for evaluating it accurately.
The general form of a cube root function is: \[ g(t) = \sqrt[3]{t} \]This expression means we are looking for a number which, when cubed, gives the original value.- **Example 1**: With \( t = 27 \), the cube root is 3 because \( 3^3 = 27 \).- **Example 2**: With \( t = -\frac{1}{8} \), the cube root is \(-\frac{1}{2}\) since \( (-\frac{1}{2})^3 = -\frac{1}{8} \).
Cubic functions, unlike square roots, can handle negative inputs and still return valid outputs, illustrating the versatility and extensive applications of cube root functions.

Substitution in Functions

Substitution in functions is a fundamental concept in algebra. It's about replacing the variable in a function with a specific value to find the output. This technique allows you to see how a particular input affects the function's result.
To perform substitution, follow these steps:

• Identify your function's variable. In our example, this is \( t \).
• Plug in the given number in place of the variable.
• Simplify the expression to determine the output.

In the original exercise, we substituted 27 and \(-\frac{1}{8}\) into the cube root function to evaluate it. This straightforward substitution process ensures we calculate the exact numerical value for each input easily, demonstrating the usefulness of substitution in functional evaluation.