Problem 8 - Calculus with Concepts in Calculus

Question

Find the numerical value of the function at the given values of \(a\). $$ g(t)=|2-t| ; a=6 $$

Step-by-Step Solution

Answer

The value of \( g(6) \) is 4.

Step 1: Substitute the value of 'a' into the function

Given the function \( g(t) = |2 - t| \), we replace \( t \) with \( a = 6 \). This gives us \( g(6) = |2 - 6| \).

Step 2: Simplify the expression inside the absolute value

Calculate the expression \( 2 - 6 \) to simplify it to \( -4 \). Therefore, we have \( g(6) = |-4| \).

Step 3: Evaluate the absolute value

The absolute value of \( -4 \) is \( 4 \), because absolute value represents the distance from zero without regard to direction. Thus, \( g(6) = 4 \).

Key Concepts

Numerical Evaluation

Numerical evaluation involves determining the value of an expression or function for specific values of its variables. It is essentially the process of calculating or finding the number that satisfies the given conditions of the problem. In the context of the provided exercise, we aim to find the numerical value of the function \(g(t) = |2-t|\) when \(t\) is given as \(6\). To do this, we follow a logical sequence of steps:

• First, substitute the provided value into the function.
• Then, simplify within the absolute value operation.
• Finally, evaluate the absolute value to get the result.

This process ensures we correctly interpret and calculate the function at the point of interest.

Substitution

Substitution is a fundamental technique in mathematics where we replace a variable with its corresponding numerical value. This is done to simplify the equation or function and to allow for further calculation. In our example, the function \(g(t) = |2-t|\) requires us to replace \(t\) with the given value of \(a = 6\). This yields the expression \(g(6) = |2 - 6|\). By substituting, you're effectively creating a scenario where the function can be evaluated at a specific, concrete point. This is a crucial step in both algebra and calculus, allowing for precise calculations and solutions.

Simplification

Simplification involves reducing an expression to its simplest form to make calculations easier. After substitution, our task is to simplify the expression inside the absolute value, which is \(2 - 6\). This results in \(-4\), leading to the simplified expression \(g(6) = |-4|\). The shift from \(2-6\) to \(-4\) is a straightforward arithmetic operation, but it is a crucial step that sets the stage for evaluating the absolute value. Simplification makes subsequent steps easier to handle, and reduces the chances of errors in calculations.

Function Evaluation

Function evaluation refers to the process of determining the output value of a function given a specific input. For absolute value functions, this means determining the non-negative distance of a number from zero. Once simplified, our expression is \(g(6) = |-4|\). The absolute value concept means we take the magnitude, or the positive form, of that number. The absolute value of \(-4\) is \(4\), which is the distance of \(-4\) from zero on the number line. Thus, the computed function value is \(g(6) = 4\). This concept of function evaluation helps in understanding how functions transform inputs into outputs, and is the essence of working with mathematical operations and expressions.