Problem 44 - College Algebra

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 \( f(5) \) is 62.

Step 1: Understand the Function and the Value of x

The given function is \( f(x) = 4(2)^{x-1} - 2 \). We need to evaluate \( f(5) \), which means we substitute \( x = 5 \) into the function.

Step 2: Substitute x with 5

Replace \( x \) with 5 in the function:\[ f(5) = 4(2)^{5-1} - 2 \]

Step 3: Simplify the Exponent

Calculate the exponent: \( 5-1 = 4 \). So, the expression becomes:\[ f(5) = 4(2)^4 - 2 \]

Step 4: Calculate the Exponential Term

Find the value of \( (2)^4 \):\( 2^4 = 16 \), thus:\[ f(5) = 4 \times 16 - 2 \]

Step 5: Perform Multiplication

Multiply 4 by 16:\[ 4 \times 16 = 64 \]Thus, our function becomes:\[ f(5) = 64 - 2 \]

Step 6: Final Calculation

Subtract 2 from 64:\[ 64 - 2 = 62 \]So, \( f(5) = 62 \).

Key Concepts

Evaluating Functions

Evaluating functions is a fundamental concept in mathematics. It involves finding the output value of a function for a specific input. Functions act like mathematical machines: you input a value, let's call it "x," and the function transforms it into an output value. For exponential functions like the one we have here, you follow these steps:

• Identify the given function and note the value of the variable for which you need to find the output.
• Substitute this variable's value into the function using correct mathematical operations.
• Calculate the result following the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

This process helps you solve expressions and assess the behavior of functions efficiently.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators like addition, subtraction, multiplication, and division. In the context of functions, understanding algebraic expressions is crucial as it helps in simplifying and calculating the required output.
For the function involved in this exercise, the expression is: \[ f(x) = 4(2)^{x-1} - 2 \]Here, "4" is a coefficient, "2" is the base of the exponential term, and "x-1" is the exponent. By substituting different values into "x," you can determine how the function behaves.
Simplifying algebraic expressions often involves

• Performing operations inside parentheses first.
• Applying exponents.
• Dealing with multiplication or division.
• Handling addition or subtraction.

You need to keep your steps clear and organized to ensure accuracy in your calculations.

Problem-Solving Steps

Solving mathematical problems systematically can make complex calculations more manageable. Following a structured series of steps ensures you do not miss any essential calculations, especially in exponential functions.
The process used in the original exercise solution follows:

• Identify and Understand the Problem: Recognize the type of function and the given values.
• Substitution: Insert the specific value of "x" into the equation.
• Simplification of Exponents: Calculate any exponents in the function.
• Perform Necessary Operations: Carry out multiplications or divisions.
• Final Calculation: Conduct any final additions or subtractions to get the outcome.

By steadily working through each step, you ensure precision and improve your understanding of the function's behavior. This approach can be applied to various problem-solving scenarios across mathematics.