Problem 23 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$y>2^{x}$$

Step-by-Step Solution

Answer

To graph the inequality \(y > 2^{x}\), plot the function \(y = 2^{x}\) and then shade the region located above the curve to represent the solution set where \(y\) is greater than \(2^{x}\).

Step 1: Grasp the inequality

The inequality is \(y > 2^{x}\), which means that the y-values we are considering are strictly greater than the corresponding y-values for the function \(f(x) = 2^{x}\).

Step 2: Graph the function

Now, graph the function \(f(x) = 2^{x}\). This function starts at the point (0, 1), decreases towards y=0 to the left of x=0 and increases exponentially to the right of x=0.

Step 3: Identify the region corresponding to the inequality

Since the inequality is \(y > 2^{x}\), the solution set will be the region above the graph of \(y=2^{x}\). Shade this region to indicate that y-values in this region are greater than \(2^{x}\) for same x-values.

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in mathematics, especially in algebra and calculus. These functions involve the growth or decay rates characterized by the constant base raised to a variable exponent. A classic example is the function \( f(x) = 2^x \), where 2 is the base and \( x \) is the exponent. This type of function is known for rapid growth.

• Base and Exponent: In \( f(x) = 2^x \), 2 is the base, and \( x \) is the exponent. The base must be a positive number for the function to be well-defined across all real numbers.
• Graph Characteristics: The graph of an exponential function \( f(x) = a^x \) crosses the y-axis at (0, 1) if there are no additional transformations like shifting or scaling. The y-values asymptotically approach zero as \( x \) goes negative, and they increase without bound as \( x \) becomes positive.
• Applications: Exponential functions model many real-world scenarios, such as population growth, radioactive decay, and interest on investments.

Understanding the behavior of exponential functions is crucial for solving problems involving growth processes and complex equations.

Inequality Solutions

Inequality solutions involve finding all values that satisfy an inequality condition. In our exercise, we deal with the inequality \( y > 2^x \). This means we are looking for y-values that are strictly greater than the corresponding values given by the exponential function.

• Inequality Sign: The 'greater than' sign (>) indicates that the solution includes all points above the curve of \( 2^x \).
• Boundary Condition: The line \( y = 2^x \) acts as a border. Points on this line do not satisfy the inequality because \( y \) must be strictly greater than \( 2^x \).
• Solution Set Representation: The region above the curve shows all solutions. This requires graphing and properly shading the region that meets the inequality's condition.

Solving inequalities means understanding how equations create boundaries and regions on a graph.

Graphing Techniques

Graphing inequalities involves several distinct steps that ensure the correct visualization of solutions. It's an essential skill for depicting and understanding the range of possible solutions visually.

• Plotting the Graph: First, graph the equation \( y = 2^x \) without any inequality. This involves plotting the characteristic exponential curve, starting from the y-intercept at (0, 1), increasing rapidly as \( x \) moves right.
• Identifying the Solution Region: In the inequality \( y > 2^x \), the solution is the area above the line. Hence, you plot the curve as a dashed line to indicate the boundary is not included, then shade the area above.
• Checking Points: To confirm correct shading, select a test point not on the curve, like (0, 2), and substitute into the inequality to verify whether it satisfies the condition.

Mastering these techniques will enhance your ability to solve and interpret inequalities graphically, crucial for many areas in math and science.