Problem 98 - A Graphical Approach to College Algebra

Question

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=2^{3 x}-8^{x-3}$$

Step-by-Step Solution

Answer

The equation has no real solutions; investigate graphically for inequalities.

Step 1: Simplify the Equation

Begin by rewriting the equation using the properties of exponents. Notice that \(8 = 2^3\), so \(8^{x-3} = (2^3)^{x-3} = 2^{3(x-3)} = 2^{3x-9}\). The equation becomes \(2^{3x} - 2^{3x-9} = 0\).

Step 2: Factor the Expression

Factor the expression \(2^{3x} - 2^{3(x-3)}\) as a difference of powers. Factor out the common term, which is the smallest power, \(2^{3(x-3)} = 2^{3x-9}\). The expression becomes \(2^{3x-9}(2^9 - 1) = 0\).

Step 3: Solve the Equation

Since the equation is \(2^{3x-9}(2^9 - 1) = 0\), and knowing that \(2^9 - 1\) is not zero, set \(2^{3x-9} = 0\). However, an exponential function never equals zero. This implies no real solutions for \(f(x) = 0\) when considering real values of \(x\).

Step 4: Analyze the Graph

Consider the properties of the function \(f(x) = 2^{3x} - 8^{x-3}\). The graph represents an exponential curve. As no real solutions satisfy \(f(x) = 0\), observe that the function remains either entirely above or below the x-axis.

Step 5: Solve the Inequality \(f(x) < 0\)

Check the behavior of the function \(f(x)\) when it is negative. Since \(2^{3(x-3)}\) becomes very small relative to \(2^{3x}\) as \(x\) increases, it suggests \(f(x)\) remains positive for large \(x\). Investigate smaller \(x\) using graph or numerical analysis to find intervals where \(f(x) < 0\).

Step 6: Solve the Inequality \(f(x) \geq 0\)

Since \(f(x)\) does not equate to zero, \(f(x) \geq 0\) essentially equates to \(f(x) > 0\). Based on the analysis from Step 5, determine intervals where the graph lies above the x-axis, ensuring you note any transitions between positivity and negativity.

Key Concepts

Inequalities

When dealing with inequalities, we're often looking into regions where a function fits certain conditions, like being greater or less than zero. In this exercise, the inequalities we need to consider are:

• \( f(x) < 0 \): The function is negative in certain intervals.
• \( f(x) \geq 0 \): The function is non-negative, meaning it is either positive or zero.

To solve these inequalities, the focus is on understanding when and where the function dips below or stays above the x-axis. With the function given, \[ f(x) = 2^{3x} - 8^{x-3} \],you can explore the regions analytically by simplifying the exponents and noting the behavior as \( x \) changes. Solving inequalities graphically, as mentioned in the original solution, involves identifying these regions on a graph of \( y = f(x) \). The key is to determine when \( 2^{3x} \) dominates over \( 8^{x-3} \), as this will inform when the function remains positive or negative.

Graphical Analysis

Graphical analysis helps to visualize the function's behavior across different intervals. When analyzing the function \( f(x) = 2^{3x} - 8^{x-3} \) using a graph:

• First, plot the function over various values of \( x \) to see where it crosses the x-axis.
• Since the analytical solution found no real roots for \( f(x) = 0 \), focus on the curve's position relative to the x-axis to identify ranges of positivity or negativity.

Understanding the behavior visually, the graph's curvature illustrates how the components \( 2^{3x} \) and \( 8^{x-3} \) interact. Analyzing test points on the graph, especially around suspected transitions, can confirm if \( f(x) \) dips below zero or remains above. Use tools like graphing calculators or software to plot the function accurately, as they make it easier to identify specific intervals where inequalities hold true. This visual method complements the algebraic analysis, offering a clearer idea of the function's overall dynamic.

Exponential Functions

Exponential functions, like the one in this exercise, are characterized by a constant base raised to a variable exponent. Here, we have:\[ f(x) = 2^{3x} - 8^{x-3} \]The properties of exponential functions are crucial in understanding how they behave:

• Exponential functions grow quickly. The rate of growth depends on the base; for instance, base 2 functions like \( 2^{3x} \) expand rapidly as \( x \) increases.
• The base of an exponential term cannot result in zero since a positive number raised to any real power remains positive.

In the case of this function, the challenge lies in simplifying terms like \( 8^{x-3} \) by remembering that \( 8 = 2^3 \), allowing the exponent to be rewritten in terms of base 2. These properties give us insights when tackling equations and inequalities, simplifying complex expressions by focusing on common bases.