Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=e^{x},\) for \(f(3)\)

Step-by-Step Solution

Answer

The value of \( f(3) = e^3 \) is approximately 20.0855.

Step 1: Identify the function and input

We are given the function \( f(x) = e^x \) and need to evaluate it at \( x = 3 \). This means we have to substitute 3 into the function in place of \( x \).

Step 2: Substitute the input into the function

Substituting \( x = 3 \) into the function \( f(x) = e^x \), we get \( f(3) = e^3 \).

Step 3: Calculate the value of \( e^3 \)

Use a calculator to evaluate \( e^3 \). The approximate value of \( e^3 \) is 20.0855 when rounded to four decimal places.

Key Concepts

Function Evaluation

Evaluating a function involves finding the output value of a function for a given input. In simpler terms, it's like seeing what the machine spits out when you feed it a particular number. For our exercise, the function is given as \( f(x) = e^x \). Here, \( e^x \) is the exponential function where \( e \) is a constant and \( x \) is a variable representing any number you input.To evaluate the given function at \( x = 3 \):

• Replace \( x \) in the function \( f(x) = e^x \) with the number 3.
• This substitution gives us \( f(3) = e^3 \).

Once substituted, you simply calculate the expression using the constant \( e \). This process helps you determine the exact outcome when 3 is the input for the function \( e^x \).

Natural Exponential Constant

The letter \( e \) in mathematics refers to the natural exponential constant, approximately equal to 2.71828. It's a magical number that appears in various areas of mathematics, especially in growth processes like compound interest, populations, or radioactive decay.This constant is known as Euler's number and serves as the base for natural logarithms. In our exercise, it's important because it forms the foundation of the given function \( f(x) = e^x \). Consider \( e^3 \):

• The exponent of 3 means we multiply \( e \) by itself three times.
• In simpler steps, imagine multiplying 2.71828 * 2.71828 * 2.71828 to get the result.

These steps show how the value of \( e^x \) grows exponentially as you increase \( x \). It's this exceptional quality of \( e \) that makes it so useful and significant in mathematics.

Rounding Decimal Places

In many calculations, especially when dealing with irrational numbers like \( e \), you need to round the result to make it more manageable and understandable. This exercise requires rounding to four decimal places to provide a precise yet straightforward answer.Following the steps in our problem, after substituting and calculating \( e^3 \), a calculator might give a number like 20.0855369. To round:

• Look at the fifth decimal place in 20.0855369, which is 3.
• Since 3 is less than 5, you round down, keeping the fourth decimal place unchanged.
• Hence, you get 20.0855 as the rounded number to four decimal places.

Rounding is a crucial mathematical skill, ensuring that results are concise, easy to read, and appropriate for practical use.