Problem 15 - Precalculus with Limits

Question

Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically to verify your approximation. \(f(x)=2^{x}\) \(g(x)=8\)

Step-by-Step Solution

Answer

The intersection point of the graphs of \(f(x)=2^{x}\) and \(g(x)=8\) is at \(x = 3\).

Step 1: Approximation

First, an approximate intersection of the two graphs can be found by drawing these two functions on the same graph. Where the two functions intersect is the approximate solution. The equation \(f(x) = 2^{x}\) is an exponential function, while \(g(x) = 8\) is a horizontal line. The intersection is approximately at \(x = 3\), as that is where the exponential curve and the horizontal line meet.

Step 2: Algebraic Verification

To verify the approximation algebraically, solve the equation \(f(x)=g(x)\), i.e. \(2^{x}=8\). To isolate \(x\), we can take the logarithm of both sides, then use the property of logs that allows us to pull down the exponent: \(\log_{2}(2^{x})=\log_{2}(8)\) which simplifies to \(x = \log_{2}(8)\). Evaluating this gives \(x = 3\), confirming our approximation.

Key Concepts

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the equation \( f(x) = 2^x \), the base, 2, is constant while \( x \), the exponent, can change. This causes the value of the function to increase rapidly as \( x \) becomes larger. Exponential functions appear frequently in real-world situations such as population growth and compound interest because they model situations where growth is proportional to the current amount.

• As \( x \) increases, \( f(x) = 2^x \) grows quickly due to its exponential nature.
• This growth pattern is what distinguishes exponential functions from linear ones, which grow at a constant rate.

Understanding the properties of exponential functions is crucial for solving equations like \( f(x) = 2^x \), especially when combined with other functions like constant lines.

Logarithms

Logarithms are the inverse operation of exponentiation. They help us solve equations where the unknown variable is an exponent, just like in the equation \( 2^x = 8 \) from the original problem. The logarithm \( \log_b(a) \) answers the question: "To what power must the base \( b \) be raised, to produce the number \( a \)?" For example, \( \log_2(8) = 3 \) because \( 2^3 = 8 \).

• Using logarithms, you can convert an exponential equation into a simpler form that is much easier to solve.
• It's important to choose the base of the logarithm that matches the base of the exponential function for straightforward calculation.

Knowing how to work with logarithms is key to finding solutions to exponential equations like \( 2^x = 8 \). They allow us to determine the exact point where two functions intersect, both graphically and algebraically.

Graph Intersection

Finding the intersection of graphs is a common problem in mathematics that involves determining where two functions have the same value for \( x \). In the context of the exercise, this means finding the point where \( f(x) = 2^x \) and \( g(x) = 8 \) meet. Graphically, this involves plotting both functions on the same coordinate plane and identifying their meeting point.

• In practice, this can often be done "by eye" to get an approximate solution.
• Graph intersection is important because it provides a visual confirmation of an algebraic solution.

Algebraically, finding the intersection involves solving the equation set by the functions being equal, i.e., determining the \( x \) value that satisfies both open equations. This is typically verified through calculations like those using logarithms, ensuring the approximation is correct.