Problem 230 - Calculus Volume 1

Question

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. $$ f(x)=(0.3)^{x} \text { a. } x=-1 \text { b. } x=4 \text { c. } x=-1.5 $$

Step-by-Step Solution

Answer

f(-1) = 3.33, f(4) = 0.0081, f(-1.5) = 6.10.

Step 1: Evaluate \( f(x) \) for \( x = -1 \)

To evaluate the function \( f(x) = (0.3)^x \) at \( x = -1 \), substitute \( -1 \) for \( x \) in the function: \[ f(-1) = (0.3)^{-1} \] To find \( (0.3)^{-1} \), we take the reciprocal of 0.3, which gives us: \[ f(-1) = \frac{1}{0.3} = 3.33 \] Thus, \( f(-1) \) is 3.33.

Step 2: Evaluate \( f(x) \) for \( x = 4 \)

Substitute \( 4 \) for \( x \) in the function: \[ f(4) = (0.3)^4 \] Calculate \( (0.3)^4 \) by multiplying 0.3 four times: \[ (0.3) \times (0.3) \times (0.3) \times (0.3) = 0.0081 \] Thus, \( f(4) \) is 0.0081.

Step 3: Evaluate \( f(x) \) for \( x = -1.5 \)

To evaluate the function \( f(x) = (0.3)^x \) at \( x = -1.5 \), we substitute \( -1.5 \) for \( x \): \[ f(-1.5) = (0.3)^{-1.5} \] First, calculate the reciprocal raised to the power 1.5: \[ (0.3)^{-1.5} = \frac{1}{(0.3)^{1.5}} \] Calculate \( (0.3)^{1.5} \) by finding the square root of cube result (\(0.3^3\)): \[ 0.3^3 = 0.027 \quad \text{so}\quad (0.3)^{1.5} = \sqrt{0.027} \approx 0.164 \] Now find the reciprocal: \[ \frac{1}{0.164} \approx 6.10 \] Thus, \( f(-1.5) \) is approximately 6.10.

Key Concepts

Function Evaluation

Evaluating a function means substituting a particular value for the variable in a function. It is the process of determining the output of a function when its input is known. In our case with the function \( f(x) = (0.3)^x \):

• When \( x = -1 \), you substitute \(-1\) into the function. This gives you \( (0.3)^{-1} \).
• When \( x = 4 \), substitute \( 4 \) into the function to calculate \( (0.3)^4 \).
• When \( x = -1.5 \), substitute \( -1.5 \) to find \( (0.3)^{-1.5} \).

Each substitution is done to obtain a specific result based on the power rules applied to 0.3. Evaluating a function is essential in mathematics as it shows how one quantity changes in relationship to another, categorized by the function chosen.

Reciprocal Calculation

The concept of reciprocal is crucial when dealing with negative exponents in mathematics. The reciprocal of a number is the number that, when multiplied by the original number, results in 1. For example, the reciprocal of 0.3 is \( \frac{1}{0.3} = 3.33 \).

In exponential functions, when you have a negative exponent, finding the reciprocal is a key step:

• For \( (0.3)^{-1} \), you calculate it as \( \frac{1}{0.3} \), turning it into a positive exponent scenario.
• Similarly, for \( (0.3)^{-1.5} \), take \( \frac{1}{(0.3)^{1.5}} \).

The reciprocal makes it easier to comprehend negative exponents by showing equivalence through division, simplifying complex calculations.

Exponent Rules

Exponent rules are fundamental in solving expressions involving powers. When dealing with exponential functions like \( f(x) = (0.3)^x \), understanding these rules will help simplify complex operations.

Key exponent rules include:

• Positive powers: Raising a number to a positive exponent involves multiplying that number by itself as many times as indicated. For example, \( (0.3)^4 \) means \( 0.3 \times 0.3 \times 0.3 \times 0.3 = 0.0081 \).
• Negative powers: A negative exponent signifies taking the reciprocal of the base and then applying the positive exponent. So, \( (0.3)^{-1} \) means \( \frac{1}{0.3} \).
• Fractional exponents: These signify roots. For instance, \( (0.3)^{1.5} \) involves calculating a power and a root expression, often leading from cubed to square root relations.

Understanding these rules enables students to break down complex exponential functions into simple, manageable steps, ensuring accuracy in calculations.