Problem 112 - Precalculus

Question

For the following exercises, evaluate the exponential functions for the indicated value of \(x .\) $$f(x)=4(2)^{x-1}-2 \text { for } f(5)$$

Step-by-Step Solution

Answer

The value of the function when \(x = 5\) is 62.

Step 1: Substitute x into the function

Start by substituting the value of \(x\) with 5 in the function \(f(x) = 4(2)^{x-1} - 2\). This gives \(f(5) = 4(2)^{5-1} - 2\).

Step 2: Simplify the exponent

Simplify the expression \(2^{5-1}\) by calculating the exponent. This results in \(2^4 = 16\).

Step 3: Calculate the multiplication

Multiply the result from the last step by 4. Compute \(4 \times 16\) to get 64.

Step 4: Subtract to complete the evaluation

Subtract 2 from 64 as per the equation: \(f(5) = 64 - 2\). This results in \(f(5) = 62\).

Key Concepts

Function Evaluation

Function evaluation involves finding the output of a function for a specific input. In the given exercise, we have the function \(f(x) = 4(2)^{x-1} - 2\), and we are asked to evaluate \(f(5)\).
To evaluate, it means substituting a given value (here 5) in place of \(x\) and then performing arithmetic operations to find the result. This is a standard procedure in algebra and is crucial in understanding how functions behave at specific points.

• First, replace the \(x\) in \(f(x)\) with 5, so the function becomes \(f(5) = 4(2)^{5-1} - 2\).
• Follow up by simplifying inside the exponent and evaluating the resulting expression step by step.

This step-by-step substitution is fundamental in analyzing mathematical functions, allowing you to see how changes in input values affect the function's output.

Exponents

Exponents play a crucial role in exponential functions. They represent how many times a number, known as the base, is multiplied by itself. In the function \(f(x) = 4(2)^{x-1} - 2\), the exponent is \(x-1\).
When evaluating exponents, it's essential to perform the operation inside the exponent first before dealing with other parts of the expression. In this example, calculating \(2^{x-1}\) is the key step. Once you've substituted \(x = 5\), the expression becomes \(2^{5-1}\), which simplifies to \(2^4\).

• Understanding zero exponents: Any non-zero base raised to the power of 0 is 1.
• Negative exponents indicate division; for instance, \(2^{-2}\) is \(1/2^2\).

Mastering exponents is essential for evaluating exponential functions accurately. It lays the groundwork for more complex math topics, such as logarithms and calculus.

Substitution Method

The substitution method is a technique used to simplify mathematical expressions and equations by replacing a variable with a given value. In this exercise, we substitute \(x = 5\) into the function \(f(x) = 4(2)^{x-1} - 2\).
The substitution method is straightforward but must be applied carefully to ensure correct results. The steps include:

• Identify the variable to substitute.
• Replace it with the given number (for instance, substituting \(x = 5\)).
• Evaluate the expression using order of operations.

This method is often the first step in solving equations and can also be used to check the solutions. It simplifies the process of dealing with complex expressions by breaking them down into manageable parts, facilitating a clearer understanding of the function's behavior.