Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-\frac{3}{2}(3)^{-x}+\frac{3}{2},\) for \(f(2)\)

Step-by-Step Solution

Answer

1.3333

Step 1: Substitute the Value of x into the Function

Given the function \(f(x) = -\frac{3}{2}(3)^{-x} + \frac{3}{2}\), we need to evaluate this at \(x = 2\). Start by substituting \(2\) for \(x\) in the function: \[f(2) = -\frac{3}{2}(3)^{-2} + \frac{3}{2}.\]

Step 2: Calculate the Exponentiation

Now, calculate \((3)^{-2}\): \((3)^{-2} = \frac{1}{3^2} = \frac{1}{9}.\)Substitute this back into the expression: \[f(2) = -\frac{3}{2} \times \frac{1}{9} + \frac{3}{2}.\]

Step 3: Perform the Multiplication

Multiply \(-\frac{3}{2}\) by \(\frac{1}{9}\): \[-\frac{3}{2} \times \frac{1}{9} = -\frac{3}{18} = -\frac{1}{6}.\]Thus the expression becomes:\[f(2) = -\frac{1}{6} + \frac{3}{2}.\]

Step 4: Perform the Addition

Combine \(-\frac{1}{6}\) with \(\frac{3}{2}\). First, express \(\frac{3}{2}\) with a common denominator of 6: \(\frac{3}{2} = \frac{9}{6}\).Now perform the addition: \[-\frac{1}{6} + \frac{9}{6} = \frac{8}{6} = \frac{4}{3}.\]

Step 5: Convert to Decimal Form and Round if Necessary

Finally, convert \(\frac{4}{3}\) to decimal form by performing the division:\(\frac{4}{3} \approx 1.3333.\)Hence, the result rounded to four decimal places is 1.3333.

Key Concepts

Substitution in Functions

When working with functions, substitution is key to evaluating them at specific points. Think of a function like a machine with an input and output. In our case, the function is given by: \[f(x) = -\frac{3}{2}(3)^{-x} + \frac{3}{2}\]To evaluate it for a specific value of \(x\), you substitute that value into the function. For example, to find \(f(2)\), simply replace every instance of \(x\) in the function with \(2\). This process helps find a particular output for the corresponding input value. It's straightforward but foundational for understanding functions.

• Identify the value to substitute (e.g., \(2\) in this case).
• Replace \(x\) with \(2\) in the expression.

After substitution, the function transforms into an expression that can be evaluated numerically.

Exponentiation Rules

Exponentiation involves raising numbers to a power. An important concept is the handling of negative exponents, which flip the number to its reciprocal. For instance, \((3)^{-2}\) is equivalent to flipping \(3^2\), giving: \[(3)^{-2} = \frac{1}{3^2} = \frac{1}{9}\]Negative exponents can often appear confusing, but the rule essentially means you are dealing with fractions. Here are some simple tips:

• A negative exponent signifies the reciprocal of the base raised to the positive of that exponent.
• Always pause to calculate the reciprocal before moving to further computations.

Understanding exponentiation rules, including negative exponents, allows for accurate function evaluation.

Fraction Arithmetic

Working with fractions involves arithmetic rules that can sometimes trip you up. In our function evaluation, we needed to multiply and add fractions. Let’s break it down:For multiplication: \[-\frac{3}{2} \times \frac{1}{9} = -\frac{3}{18} = -\frac{1}{6}\]Simple steps to remember:

• Multiply the numerators with each other.
• Multiply the denominators with each other.
• If possible, simplify the fraction by dividing common factors.

For addition:Adding \(-\frac{1}{6}\) and \(\frac{9}{6}\):

• Ensure both fractions have the same denominator.
• Combine the numerators, keeping the denominator the same: \(-\frac{1}{6} + \frac{9}{6} = \frac{8}{6}\).
• Simplify \(\frac{8}{6}\) to get \(\frac{4}{3}\).

Mastering these steps makes fraction arithmetic a breeze.

Decimal Conversion

After dealing with fractions, sometimes you need to convert them into decimals, especially when rounding is necessary. The goal is to express the result in an easy-to-read form. For \(\frac{4}{3}\), division yields:\[\frac{4}{3} = 1.3333\]This conversion is useful in various scenarios, like making calculations consistent. Important points include:

• Perform division on the numerator by the denominator.
• When instructed, round to a certain number of decimal places to increase precision or simplify.

In tasks like these, rounding to four decimal places (1.3333 here) is often necessary for clarity and precision.