Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-2 e^{x-1},\) for \(f(-1)\)

Step-by-Step Solution

Answer

The value of \( f(-1) \) is approximately \(-0.2706\).

Step 1: Substitute the value into the function

To find \( f(-1) \), substitute \(-1\) in place of \(x\) in the function \( f(x) = -2e^{x-1} \). This gives us \( f(-1) = -2e^{-1-1} \).

Step 2: Simplify the expression in the exponent

Simplify the expression in the exponent: \(-1 - 1\) becomes \(-2\). Thus, the function becomes \( f(-1) = -2e^{-2} \).

Step 3: Evaluate the exponential expression

Use the exponential function to evaluate \( e^{-2} \). The value of \( e^{-2} \) is approximately \( 0.1353 \) when rounded to four decimal places.

Step 4: Calculate the final result

Multiply \(-2\) by the approximate value of \( e^{-2} \): \( f(-1) = -2 \times 0.1353 \). This results in \( -0.2706 \).

Step 5: Round the answer

Since the multiplication in the previous step already results in the value being rounded to four decimal places, no further rounding is needed.

Key Concepts

Function Evaluation

Evaluating a function means replacing every instance of the variable, typically named \( x \), with a specific value from its domain and then performing the necessary arithmetic operations to find the outcome. This core idea lets us determine the value a function outputs for a given input. Let's take a deeper dive into the example function \( f(x) = -2e^{x-1} \). In this case, when asked to evaluate \( f(-1) \), we substitute \(-1\) in place of \( x \). This substitution turns the function into \( f(-1) = -2e^{-1-1} \). Making substitutions correctly and simplifying within a function is the foundational step in function evaluation.

Key points about function evaluation include:

• Substitute the input value wherever the variable appears in the function.
• Perform arithmetic operations as dictated by the expression.
• Simplify the result according to mathematical operations and rules.

Function evaluation helps in understanding how changes in the input (\( x \)) affect the output. It's especially useful when analyzing patterns and behaviors of functions.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. In the function \( f(x) = -2e^{x-1} \), the exponent \( x-1 \) represents the power to which the number \( e \) is raised. Studying exponents is crucial in exponential functions, as they define the rapid increase or decrease in the function's values.

It's key to remember that:

• An exponent of 2 (\( x^2 \)) means multiplying the base by itself: \( x \times x \).
• Negative exponents, like \( e^{-2} \), equate to the reciprocal of the base raised to the positive exponent: \( \frac{1}{e^2} \).
• When simplifying expressions with exponents, follow the order of operations carefully: exponents, then multiplication and division, and finally addition and subtraction.

In the evaluation of \( f(-1) \), simplifying the exponent \(-1 - 1\) becomes \(-2\), making our expression \(-2e^{-2}\). Understanding these principles empowers you to handle various exponential equations and expressions with confidence.

Rounding Numbers

Rounding numbers is a method of adjusting numbers to make them simpler and easier to work with. It is especially useful when dealing with long decimal places, and in ensuring answers are as specified in terms of precision.

When evaluating our function \( f(x) = -2e^{x-1} \) for \( f(-1) \), the intermediate result is \( e^{-2} \), approximately \( 0.1353352832 \), which needs to be rounded to four decimal places for simplicity. This results in \( 0.1353 \).

Essential aspects of rounding include:

• Looking at the first number to the right of your desired decimal place – if it's 5 or higher, round up.
• If it's 4 or lower, round down, leaving the last desired number unchanged.
• Always check the rounding requirements of your task – here, the requirement was to four decimal places, giving our final rounded figure of \(-0.2706\).

Rounding helps make numbers manageable and aligns calculations with the required precision level, critical in exact sciences and daily applications alike.