Problem 43 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

\( g(6) \approx 800.33 \).

Step 1: Identify the given exponential function

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

Step 2: Substitute the value of x into the function

Substitute \( x = 6 \) into the function. This gives us the expression \( g(6) = \frac{1}{3}(7)^{6-2} \).

Step 3: Simplify the exponent

Simplify the exponent in the expression. The exponent \( 6 - 2 \) calculates to \( 4 \), so the expression becomes \( g(6) = \frac{1}{3}(7)^4 \).

Step 4: Calculate \( 7^4 \)

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

Step 5: Compute the final result

Divide \( 2401 \) by \( 3 \) to find \( g(6) \). The calculation is \( g(6) = \frac{2401}{3} = 800.3333 \). Therefore, \( g(6) \approx 800.33 \).

Key Concepts

Evaluate Exponential Functions

Exponential functions are mathematical expressions where the variable is an exponent. Evaluating these functions involves calculating the output for specific input values. For example, in the function \( g(x) = \frac{1}{3}(7)^{x-2} \), we need to find what \( g(x) \) equals when \( x = 6 \). Let's go through the steps to evaluate it:

• First, substitute the given value of \( x \). Here, we substitute \( 6 \) into the function.
• This substitution yields the expression \( g(6) = \frac{1}{3}(7)^{6-2} \).
• The expression involves an exponent, which we need to simplify next.

Evaluating exponential functions helps us understand how quantities grow or decay rapidly, such as population growth or radioactive decay. It's essential to follow each step precisely to prevent small mistakes, which can lead to significantly different results.

Substitution in Functions

Substitution is the method of replacing the variable in a function with a designated number. For our function, \( g(x) = \frac{1}{3}(7)^{x-2} \), we substitute \( x = 6 \).

• The substitution step transforms our function into \( g(6) = \frac{1}{3}(7)^{6-2} \).
• This results in a straightforward expression where we can now focus on simplifying the exponent.

Substitution is crucial because it allows us to compute definitive values from otherwise abstract expressions. This process is the key to applying mathematical models to real-world scenarios. Whether solving for a specific value of \( x \) in a graph or plugging in measurements for a physics equation, substitution simplifies this task, making the math direct and applicable.

Simplifying Exponents

When working with exponents, simplifying them is a crucial step. An exponent dictates how many times the base is multiplied by itself. Let's simplify the exponent in \( g(6) = \frac{1}{3}(7)^{6-2} \).

• The exponent \( 6-2 \) simplifies to \( 4 \), turning our expression into \( g(6) = \frac{1}{3}(7)^4 \).
• Next, calculate \( 7^4 \) by multiplying: \( 7 \times 7 \times 7 \times 7 = 2401 \).
• Finally, divide \( 2401 \) by \( 3 \) to get the result \( 800.3333 \).

This result highlights the importance of clear exponent handling. Mistaking or skipping steps in simplifying exponents can lead to wrong answers. Understanding exponents also extends beyond this exercise; it forms the foundation for more advanced topics such as logarithms and powers of complex numbers.