Problem 38 - Precalculus Mathematics for Calculus

Question

(a) Sketch the graphs of \(f(x)=9^{x / 2}\) and \(g(x)=3^{x}\) (b) Use the Laws of Exponents to explain the relationship between these graphs.

Step-by-Step Solution

Answer

The graphs are identical since \(f(x) = 9^{x/2}\) simplifies to \(g(x) = 3^x\).

Step 1: Understanding the Functions

We need to sketch the graphs for two different functions: \(f(x)=9^{x/2}\) and \(g(x)=3^x\). Recognize that \(9^{x/2}\) can be rewritten using the Laws of Exponents.

Step 2: Rewriting the Function

Recognize that \(9\) can be expressed as \(3^2\). Thus, \(f(x)=9^{x/2}\) can be rewritten as \(f(x)=(3^2)^{x/2}\). According to the Law of Exponents \((a^m)^n = a^{mn}\), this simplifies to \(f(x)=3^x\).

Step 3: Identifying the Relationship

Since \(f(x)=3^x\) and \(g(x)=3^x\), both functions are identical. Therefore, the graphs of \(f(x)\) and \(g(x)\) are the same.

Step 4: Graph Sketching

Plot the graph of \(h(x) = 3^x\) for reference. Since both functions are equivalent, sketch a single graph that represents both \(f(x)\) and \(g(x)\), which is an exponential growth curve starting from \( (0, 1) \) and increasing as \(x\) increases.

Key Concepts

Graphs of Exponential Functions

Graphs of exponential functions are fascinating and tend to capture the interest of many students due to their unique shape and behavior. An exponential function will generally have the form \(f(x) = a^x\), where \(a > 0\) and \(a eq 1\). These functions have the remarkable property of exponential growth or decay, depending on the base \(a\). Let’s break down what a typical graph of an exponential function looks like.

• The graph passes through the point \((0, 1)\) because \(a^0 = 1\). This is true for any exponential function when \(a > 0\).
• If \(a > 1\), the function represents exponential growth. The graph will rise steeply, especially as \(x\) becomes larger. It gets closer to the y-axis as \(x\) becomes negative but never touches it. This means there is a horizontal asymptote along the x-axis.
• When \(0 < a < 1\), the function represents exponential decay. The graph will fall steeply as \(x\) increases.

In our exercise, both functions \(f(x)=9^{x/2}\) and \(g(x)=3^x\) eventually simplify to \(h(x) = 3^x\), representing exponential growth. Their graph will be a single curve that rapidly increases as \(x\) grows, embodying the core properties of exponential functions.

Laws of Exponents

The Laws of Exponents are critical rules that allow us to simplify expressions involving powers. They are tools that can transform complicated exponentials into more manageable forms. Here are a few key rules of exponents that every student should be familiar with:

• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
• Power of a Power Rule: \((a^m)^n = a^{mn}\)
• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Product Rule: \((ab)^n = a^n \times b^n\)

In the given exercise, we use the "Power of a Power" rule. We simplified \(9^{x/2}\) by expressing 9 as \(3^2\), and then applying the rule: \((3^2)^{x/2} = 3^{2 \times (x/2)} = 3^x\). Thanks to this rule, we’ve shown how the function \(f(x)=9^{x/2}\) is essentially the same as \(g(x)=3^x\). Mastering these laws allows students to navigate through more complex equations with confidence and ease.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, which is often represented by an exponential function \(f(x) = a^x\) where \(a > 1\). Many phenomena in real life, like population growth, compound interest, and certain bacterial growth, follow this pattern. Here's a simple explanation of exponential growth:

• As defined, when the base \(a > 1\), the function describes exponential growth.
• The rate of growth is proportional to the current value, meaning it accelerates as the quantity increases.
• The graph of an exponential growth function will curve upwards, showing that as \(x\) increases, \(f(x)\) increases significantly faster.

In the exercise, the function \(g(x) = 3^x\) demonstrates exponential growth. The base \(a = 3\) is greater than 1, indicating that each step increase in \(x\) leads to a value three times larger than where it previously was. The visual graph is thus a steeply rising curve, highlighting the potential for rapid increases. Understanding exponential growth is crucial for grasping advanced mathematical concepts and real-world applications.