Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explair= 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 because the function \(f(x)=\left(\frac{1}{2}\right)^{x}\) accurately models the process of continuously reducing the size of the image by 50%.

Step 1: Understand the exponential function

In the exponential function \(f(x)=\left(\frac{1}{2}\right)^{x}\), \(x\) represents an exponent and \(f(x)\) represents the result after applying the function. This kind of functions are mostly used to model processes that decrease or increase exponentially.

Step 2: Relate function to problem statement

When \(x=1\), \(f(x)=\left(\frac{1}{2}\right)^{1}=\frac{1}{2}\), hence the size of the image is reduced by 50%. When \(x=2\), \(f(x)=\left(\frac{1}{2}\right)^{2}=\frac{1}{4}\), hence the size of the image is again reduced by 50% from the original, to 25%. This pattern continues for further reductions.

Step 3: Conclude if the statement makes sense or not

Since the function \(f(x)=\left(\frac{1}{2}\right)^{x}\) accurately captures the process of sequentially reducing an image size by 50%, the statement makes sense.

Key Concepts

Image Reduction

Image reduction, often used in photography and digital media, involves decreasing the size of an image. This is typically measured as a percentage or fraction of the original. For instance, when reducing an image by 50%, it means resizing it to half of its previous dimensions.

In mathematical terms, this reduction is modeled by exponential functions. Every subsequent reduction will yield a smaller image. If you start with an image that is 100% of its size, reducing it repeatedly by 50% means that you're shrinking the image size exponentially. With each reduction step, the image is becoming half of its previous size. Hence, its size can be expressed via the exponential function which provides a clear mechanism for modeling such resizing tasks.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to simulate real-world scenarios. It allows us to predict and understand complex phenomena using mathematical expressions. In the context of the exercise, the exponential function is used to model the reduction in image size.

When speaking about modeling a process like image reduction, the function becomes a powerful tool. Imagine a scenario where you keep pressing the 'reduce by 50%' button on a photocopier. The function \(f(x) = \left(\frac{1}{2}\right)^{x}\) helps us predict the image size after each reduction. Here, \(x\) indicates the number of reductions performed, and the function outputs how much of the original size remains. Thus, through mathematical modeling, one can effectively represent and study such continuous reduction processes.

Exponents

Exponents play a crucial role in exponential functions, serving as the backbone for modeling processes that involve repeated multiplication or division. In the function \(f(x)=\left(\frac{1}{2}\right)^{x}\), \(x\) is the exponent that dictates how many times the base (\(\frac{1}{2}\)) is multiplied by itself.

Understanding exponents is vital in interpreting exponential functions like the one used in image reduction. When \(x\) is 1, the function results in \(\frac{1}{2}\), indicating an initial 50% size. With \(x\) as 2, the result is \(\frac{1}{4}\), further reducing the image to 25% of the original. The pattern can be extended to any number \(x\), allowing us to clearly see the process of exponential diminishment in size.

Therefore, mastering the concept of exponents is essential for delving into exponential functions and comprehending how they model various scenarios, such as geometric decay in image resizing by a constant percentage.