Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to the nearest thousandth. Value \(x=9.2\) \(x=-\frac{3}{4}\) \(x=0.02\) \(x=200\) Function \(h(x)=-5.5 e^{-x}\)

Step-by-Step Solution

Answer

The evaluated results for the function \(h(x)=-5.5 e^{-x}\) with values \(x=9.2\), \(x=-\frac{3}{4}\), \(x=0.02\), and \(x=200\) are -0.000, -2.659, -5.419, and -0.000, respectively.

Step 1: Evaluate the function at \(x=9.2\)

To accomplish this, input \(x=9.2\) into the function so it becomes \(h(9.2)=-5.5 e^{-9.2}\). Use your calculator to compute this value, and round it to the nearest thousandth. The result should be -0.000.

Step 2: Evaluate the function at \(x=-\frac{3}{4}\)

Input the value \(-\frac{3}{4}\) into the function, which gives \(h(-\frac{3}{4})=-5.5 e^{-(-\frac{3}{4})}\). Compute this value using your calculator and round to the nearest thousandth. The result should be -2.659.

Step 3: Evaluate the function at \(x=0.02\)

Input \(x=0.02\) into the function, which becomes \(h(0.02)=-5.5 e^{-0.02}\). Compute this value using your calculator and round to the nearest thousandth. The computed result should be -5.419.

Step 4: Evaluate the function at \(x=200\)

Input \(x=200\) into the function, so it becomes \(h(200)=-5.5 e^{-200}\). Compute this value using your calculator and round to the nearest thousandth. The result should be -0.000.

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input. In other words, it is about substituting a given value into a function to find the result.
For the function in this exercise, you are dealing with an exponential function: \(h(x) = -5.5e^{-x}\). Here, "\(x\)" is the variable or input value and "\(h(x)\)" represents the output of the function. The exponential part of the function involves the constant \(e\), which is approximately 2.718, a number known for its natural exponential growth properties.
To evaluate the function, follow these steps:

• Substitute the given value of \(x\) into the function. Replace every "\(x\)" in the function equation with the specific numeric value.
• Perform any arithmetic operations necessary, including the calculation of the exponential \(e^{-x}\).
• Compute the entire expression using appropriate mathematical techniques or tools, typically a calculator.
• Round the computed result to the required precision, such as to the nearest thousandth in this exercise.

This method applies to any function evaluation task, supporting better understanding when approaching similar problems.

Exponential Functions

Exponential functions are a special class of mathematical functions where a constant base is raised to a variable exponent, reflecting exponential growth or decay. The base is commonly the constant \(e\), approximately 2.718, known as Euler's number. It's the limit of the expression \( (1 + 1/n)^n \) as \( n \) approaches infinity.
In an exponential function format like \(f(x) = ae^{bx}\), the constant \(a\) is the coefficient influencing the scaling of the function, while "\(b\)" determines the rate of growth or decay. When \(b\) is positive, the function shows growth, and when \(b\) is negative, it shows decay.

• An important characteristic of exponential functions is their rapid increase or decrease, making them powerful for modeling real-world scenarios such as population growth, radioactive decay, or financial interest calculations.
• The function \(h(x) = -5.5 e^{-x}\) used in the original exercise decreases as \(x\) increases since the exponent of \(e\) is negative.
• This type of function tends towards zero as \(x\) gets larger but never becomes exactly zero. It's asymptotically approaching the x-axis.

Understanding exponential functions helps in evaluating problems and recognizing patterns in algebra and calculus.

Using Calculators in Algebra

Calculators are vital tools in algebra, particularly when evaluating complex functions like the one in this exercise. They allow for quick and accurate computation of expressions that might be time-consuming or prone to error if done manually.
To effectively use a calculator for solving algebraic functions, especially with exponential terms, consider the following tips:

• Familiarize yourself with the specific model of your calculator, particularly how to input expressions involving exponentials, such as \(e^x\) functions.
• Adopt systematic input procedures: Enter expressions as they appear. Understanding parentheses and operator rules in calculator syntax can help avoid common mistakes.
• Double-check calculations by re-entering expressions if needed. It's easy to miss a small digit or sign, leading to incorrect output.
• Round answers according to the problem's requirements, such as to the nearest thousandth here, to ensure precision.

Utilizing calculators effectively has become an integral part of modern education. Calculators support students in focusing on understanding the principles behind algebraic manipulation rather than struggling with arithmetic details.