Problem 44 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The value of \(f(5)\) is 62.

Step 1: Substitute x with 5

Substitute the value of 5 for x in the function: \(f(x)=4(2)^{x-1}-2\) becomes \(f(5)=4(2)^{5-1}-2\).

Step 2: Simplify the Exponent

Calculate the exponent: \(5-1 = 4\). The function now reads: \(f(5)=4(2)^4-2\).

Step 3: Evaluate the Exponential

Calculate \(2^4\): \(2^4 = 16\). The function is now \(f(5)=4(16)-2\).

Step 4: Perform Multiplication

Multiply 4 by 16: \(4 \times 16 = 64\). The equation now reads: \(f(5)=64-2\).

Step 5: Subtract Final Results

Complete the subtraction: \(64-2 = 62\). Thus, \(f(5)=62\).

Key Concepts

Evaluating Exponential Expressions

Evaluating exponential expressions involves solving expressions that contain exponents. In our given function, we're dealing with an exponential expression of the form \(2^{x-1}\). This means we need to evaluate the base, 2, raised to the power of \((x-1)\).

An exponent tells us how many times to multiply the base by itself. For example, \(2^4\) is just 2 multiplied by itself 4 times. So, \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).

When evaluating such expressions, follow these steps:

• Identify the base and the exponent in your expression.
• Subtract or simplify any numbers in the exponent, if necessary.
• Multiply the base by itself as guided by the exponent.

Grasping these basic rules helps in evaluating more complex expressions effectively.

Functional Notation

Functional notation involves using functions instead of just plain expressions. A function, like our \(f(x) = 4(2)^{x-1} - 2\), describes a relationship between inputs (\(x\)) and outputs (\(f(x)\)).

This notation helps compactly represent complex relationships and allows us to evaluate specific inputs easily. When the problem asks for \(f(5)\), it means we need to solve the equation using \(x=5\) as the input.

Steps to use functional notation:

• Understand that \(f(x)\) indicates a function named \(f\) with \(x\) as the input variable.
• Plug in the given input value (like 5 in \(f(5)\)) into the function instead of \(x\).
• Follow through with algebraic simplifications to find the output.

By mastering functional notation, we can easily interpret and solve complex problems using just a simple input-output relationship.

Step-by-Step Solution

To successfully solve exponential functions or any complex problem, following a step-by-step approach is vital. Here's why each step matters in our solution for \(f(x) = 4(2)^{x-1} - 2\) when \(x = 5\).

Step 1: **Substitute \(x\) with 5** - Begin by replacing \(x\) in the function with the specific value you are solving for. This personalizes the equation for that specific scenario.

Step 2: **Simplify the Exponent** - Calculate \(x-1\), which is the new exponent for the base. This step makes the problem manageable by transforming it into a simple calculation.

Step 3: **Evaluate the Exponential** - Solve the exponent expression, \(2^4\), to get 16. This helps in breaking down the mathematical complexity into simple arithmetic.

Step 4: **Perform Multiplication** - Multiply the result with any coefficients, such as multiplying 16 by 4 to get 64. This adjusts the scale of the result to match the function's structure.

Step 5: **Subtract Final Results** - After all arithmetic operations, subtract any constants. For example, subtracting 2 from 64 gives the final answer, 62.

This sequential method breaks down problems into small, workable pieces which lead to an accurate solution.