Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It Graph \(y=3^{x}\) and \(y=4^{x}\). Use the graph to solve the inequality \(3^{x} < 4^{x}\)

Step-by-Step Solution

Answer

The solution to the inequality \(3^{x} < 4^{x}\) is \(x\) > 0. This is based on the points where the graph of \(3^{x}\) is below the graph of \(4^{x}\)

Step 1: Graphing \(y=3^{x}\)

Take a few values for \(x\) (-2, -1, 0, 1, 2, for instance) and compute the corresponding \(y\) values for the function \(y=3^{x}\). Therefore, using those set of points, draw the graph for the function \(y=3^{x}\) on the same set of axes.

Step 2: Graphing \(y=4^{x}\)

Similarly, take the same values for \(x\) (-2, -1, 0, 1, 2) and compute the corresponding \(y\) values for the function \(y=4^{x}\). Use these points to draw the graph for the function \(y=4^{x}\) on the same set of axes.

Step 3: Analyze The Graphs

Observe the overlapping areas, or if any line is consistently above the other.

Step 4: Solve the inequality

From the graph, we can observe that the graph of \(y=3^{x}\) is below the graph of \(y=4^{x}\) for all \(x\) > 0. Hence, inequality \(3^{x} < 4^{x}\) holds true for all \(x\) > 0.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions is a crucial step in understanding the behavior of these mathematical expressions. For functions like \( y = 3^x \) and \( y = 4^x \), the graph will show how quickly the values increase as \( x \) gets larger. To begin, start by choosing a few key values for \( x \), such as -2, -1, 0, 1, and 2. Calculating the corresponding \( y \) values helps to plot the functions accurately. For \( y = 3^x \), when \( x = -2 \), \( y \) is approximately 0.11. Similarly, compute other pairs: for \( x = 0 \), \( y = 1 \); for \( x = 2 \), \( y = 9 \). These points, once plotted on a coordinate system, create the curve representing \( y = 3^x \).

Repeat these steps for \( y=4^x \). For instance, when \( x = -2 \), \( y \) is approximately 0.06. These calculated points should be marked on the same graph to facilitate comparison. Using these visualizations, students can easily see how the \( y \)-values of each function grow and change as \( x \) varies.

Inequality Solutions

Solving inequalities involving exponential functions requires graph analysis. Using the graphs of \( y=3^x \) and \( y=4^x \) can quickly provide insight into their relationships. The inequality \( 3^x < 4^x \) indicates comparing the two exponential functions over a range of \( x \) values.

From the graph, determine where \( y = 3^x \) is less than \( y = 4^x \). This occurs where the graph of \( y=3^x \) lies below \( y=4^x \) on the coordinate plane. By close inspection, it's clear this holds true for \( x > 0 \). At \( x = 0 \), both functions have \( y = 1 \) and then \( y = 4^x \) rises more steeply thereafter, maintaining its dominance in terms of \( y \) values as \( x \) increases.

This graphical method is particularly helpful for students who better understand concepts through visual demonstration. Always return to confirm by checking a few algebraic inequalities to ensure accuracy.

Analyzing Graphs

Analyzing graphs involves more than just plotting points; it's about understanding the relationships and behaviors exhibited by the functions. By examining the graph of \( y=3^x \) and \( y=4^x \), trends become indefinitely more apparent. You can conclude which function increases faster than the other based on the steepness of their curves.

When analyzing, observe where the two graphs intersect. This is the point where both functions have the same value for \( y \). For \( y=3^x \) and \( y=4^x \), this is at \( x = 0 \). Beyond this point on the positive \( x \)-axis, \( y=4^x \) surpasses \( y=3^x \), revealing the solution to our inequality as \( x > 0 \).

Helpful tips include: \

• Looking at trends—note how quickly \( y = 4^x \) escalates compared to \( y = 3^x \).
• Identifying any intersection points, such as where both functions equal 1.
• Recognizing potential asymptotic behavior as \( x \) becomes very large or very small.

In conclusion, analyzing exponential function graphs provides a comprehensive understanding not only of the inequality but also the mathematical relationships that dictate the behavior of exponential expressions.