Problem 43 - College Algebra

Question

For the following exercises, evaluate the exponential functions for the indicated value of \(x\). $$ g(x)=\frac{1}{3}(7)^{x-2} \text { for } g(6) $$

Step-by-Step Solution

Answer

The value of \( g(6) \) is approximately 800.3333.

Step 1: Identify the Given Function

The function provided is \( g(x) = \frac{1}{3} (7)^{x-2} \). We need to evaluate this function at \( x = 6 \).

Step 2: Substitute x with 6

Replace \( x \) in the function \( g(x) = \frac{1}{3} (7)^{x-2} \) with 6, resulting in \( g(6) = \frac{1}{3} (7)^{6-2} \).

Step 3: Simplify the Exponent

Calculate the expression in the exponent: \( 6 - 2 = 4 \). Now the function is \( g(6) = \frac{1}{3} (7)^4 \).

Step 4: Evaluate the Exponential Function

Calculate \( 7^4 \), which is \( 7 \times 7 \times 7 \times 7 = 2401 \). Thus, the function becomes \( g(6) = \frac{1}{3} \times 2401 \).

Step 5: Divide by 3

Divide 2401 by 3: \( \frac{2401}{3} = 800.3333 \). So, \( g(6) = 800.3333 \).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the value of a function for a specific input. In our exercise, we have the exponential function \( g(x) = \frac{1}{3}(7)^{x-2} \), which we need to evaluate when \( x = 6 \).

• Step 1 involves identifying the function and the value at which it will be evaluated, which is "6" in this case.
• Step 2 is to substitute "6" into the function wherever you see "x". This substitution transforms the function to \( g(6) = \frac{1}{3} (7)^{6-2} \).

Substituting involves replacing the variable with the given number to find out how the function behaves at that specific point.
This step is essential as it sets up the remainder of the problem for simplification and solving.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. They are an essential tool in algebra, especially when working with exponential functions.
For example, in the equation \( 7^4 \), the base "7" is multiplied by itself "4" times.

• In this exercise, you found the exponent part of the function by simplifying \( 6 - 2 \) to get the exponent "4".
• So, \((7)^{6-2}\) becomes \( 7^4 \).

Calculating the value of \( 7^4 \) involves multiplying 7 by itself four times: \( 7 \times 7 \times 7 \times 7 = 2401 \).
Understanding how to calculate exponents helps solve complex problems that rely on repeated multiplication, such as exponential growth models.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to solve equations. This skill is vital in evaluating functions.
In the given exercise, once the exponent was simplified to \( 7^4 \), the next step was to apply algebraic techniques to simplify the entire function:

• You start with \( g(6) = \frac{1}{3} \times 2401 \).
• This step requires distributing the division across any operations connected to it to simplify the expression's value accurately.

Dividing \( 2401 \) by \( 3 \) gives \( 800.3333 \), completing the evaluation of the function.
Algebraic manipulation is crucial for isolating terms and simplifying expressions, ensuring that each step in a math problem is correct and leading to the right solution. It is used extensively in both solving equations and in evaluating complex functions.