Problem 84 - College Algebra

Question

Make Sense? In Exercises \(83-86,\) determine whether catch statement makes sense or does not make sense, and explain your reasoning. I'm using a photocopier to reduce an image over and over by \(50 \%,\) so the exponential function \(f(x)-\left(\frac{1}{2}\right)^{x}\) models the new image size, where \(x\) is the number of reductions.

Step-by-Step Solution

Answer

The statement makes sense. The function \(f(x) = \left(\frac{1}{2}\right)^x \) correctly models the scenario of reducing an image's size by \(50 \%\), where \(x\) is the number of reductions.

Step 1: Understand Exponential Decay

The function \(f(x) = \left(\frac{1}{2}\right)^x\) is a classic example of exponential decay. In this particular case, every time \(x\) (the number of reductions) increases by 1, the output of the function halves, representing a \(50 \%\) reduction.

Step 2: Apply Function to Real-Life Situation

Now apply this understanding to a photocopier reducing an image's size. With each reduction, the image is halved, or reduced by \(50 \% \). So if \(x\) is the number of reductions, then after \(x\) reductions, the image is \(\left(\frac{1}{2}\right)^x\) of its original size.

Step 3: Evaluate the Statement

Given above, the statement 'I'm using a photocopier to reduce an image over and over by \(50 \% \), so the exponential function \(f(x) = \left(\frac{1}{2}\right)^x \) models the new image' is correct. The function effectively models the situation of continuously reducing an image by \(50 \% \).

Key Concepts

Mathematical Modeling

In everyday life, we often encounter situations where certain values consistently decrease or increase over time. Mathematical modeling helps us describe these real-world occurrences using mathematical expressions and functions. Consider the instance of using a photocopier to repeatedly reduce the size of an image. Here, each action reduces the image size by half.

Mathematically, this scenario can be represented using an exponential decay model. This model uses an exponential function to express how the size of the image changes with each photocopying action. By utilizing the concept of mathematical modeling, we convert an everyday process into a mathematical problem we can analyze and solve.

• It allows predicting future behavior of the system (like how small the image will become).
• Models provide insights into how processes behave over time.
• They are foundational in engineering, economics, and even biology for predicting outcomes.

Exponential Functions

Exponential functions are a distinct category of mathematical functions. They describe processes where a quantity changes at a rate proportional to its current value. This type of function is particularly known for modeling both growth and decay scenarios effectively. For instance, in the function form \(f(x) = a^x\), where \(a\) is a constant greater than zero, the function can model situations that either grow or decrease over time. In our photocopy example, the function \(f(x) = \left(\frac{1}{2}\right)^x\) signifies exponential decay.

• The "half" base \(\frac{1}{2}\) signifies a reduction process, suitable for decay.
• Such a function halves the output each time \(x\), number of reductions, increases by 1.
• Exponential functions are versatile, fitting a variety of contexts, from population growth to nuclear decay.

Function Evaluation

Evaluating a function simply means finding the output of the function for a given input. In the context of our photocopying scenario, if we want to understand how small our image becomes after several reductions, we evaluate the exponential function. Given the function \(f(x) = \left(\frac{1}{2}\right)^x\), for any integer value of \(x\) (the number of reductions), we can determine the image's size relative to its original size.

Consider:

• If \(x = 0\), the image remains its full size as no reductions have occurred, thus \(f(0) = 1\).
• For \(x = 1\), the image is halved, so \(f(1) = \frac{1}{2}\).
• If \(x = 2\), the image becomes one-fourth of the original size, hence \(f(2) = \frac{1}{4}\).

By continuing to evaluate with increasing \(x\) values, we better understand the function's behavior and impact of repeated reductions.