Problem 30 - Precalculus with Calculus Previews

Question

Use a graph to solve the given inequality. $$ e^{x} \leq 1 $$

Step-by-Step Solution

Answer

The solution is \( x \leq 0 \) or \((-∞, 0]\).

Step 1: Understand the Inequality

The inequality we need to solve is \( e^x \leq 1 \). We want to find the values of \( x \) for which the exponential function \( e^x \) does not exceed 1.

Step 2: Analyze the Function Graphically

To solve the inequality graphically, we'll examine the graph of the function \( f(x) = e^x \) and the constant line \( y = 1 \). The inequality \( e^x \leq 1 \) holds where the graph of \( f(x) = e^x \) lies beneath or touches the line \( y = 1 \).

Step 3: Draw the Graph

Plot the graph of \( f(x) = e^x \). This is an exponential curve that passes through the point \((0,1)\) and increases rapidly for \( x > 0 \). Draw the horizontal line \( y = 1 \) across the graph.

Step 4: Identify the Solution Region

Find where the curve \( f(x) = e^x \) is below or intersects the line \( y = 1 \). The curve \( e^x \) equals 1 at \( x = 0 \) and decreases towards the x-axis (but never crosses it) for \( x < 0 \). Therefore, the solution to \( e^x \leq 1 \) is \( x \leq 0 \).

Step 5: Write the Solution

The inequality \( e^x \leq 1 \) is satisfied for all \( x \leq 0 \), meaning the set of solutions is all values from -∞ to 0, inclusive. In interval notation, this is \( (-fty, 0] \).

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics, often represented as \( f(x) = a^{x} \), where \( a \) is a positive constant. A common base for exponential functions is \( e \), known as Euler's number, approximately equal to 2.718.
This unique constant exhibits continuous growth and models real-world phenomena such as population growth and radioactive decay.

• The exponential function \( e^x \) rises sharply. It continuously increases and never turns back, approaching infinity as \( x \) becomes larger.
• At \( x = 0 \), \( e^x \) equals 1, which acts as a crucial reference point.
• The horizontal asymptote of \( e^x \) is the line \( y = 0 \), a boundary it approaches but never touches for \( x < 0 \).

Understanding exponential functions is vital as they behave quite distinctively compared to linear or polynomial functions. They hold key importance in calculus and mathematical modeling.

Graph Interpretation

Graph interpretation involves analyzing the relationships and patterns represented visually. In solving \( e^x \leq 1 \), this involves recognizing where the exponential graph meets or falls below a specific line. Interpreting graphs allows us to visualize how functions behave over intervals.
When considering the graph of \( e^x \), observe the following:

• The exponential graph passes through the point (0,1), signifying that \( x = 0 \) is where \( e^x = 1 \).
• For \( x > 0 \), the graph climbs rapidly, showing it's always above 1.
• For \( x < 0 \), the graph heads downwards but stays above 0, indicating \( e^x \) is a fraction less than 1.

By using a visual representation, solving inequalities can become more intuitive. This helps us not only solve but also understand the behavior of functions across the domain.

Inequalities in Mathematics

Inequalities are mathematical statements that involve expressions that are not exactly equal. These are crucial in comparing values and analyzing different conditions. An inequality like \( e^x \leq 1 \) dictates that the value of the exponential function should either equal or be less than 1.
Working with inequalities, one should remember:

• Look for where the expression equals the boundary (\( e^x = 1 \) at \( x = 0 \)).
• Identify regions that satisfy the inequality condition (here, \( x \leq 0 \)).
• Understand how inequality signs affect the solution set. "\( \leq \)" includes the boundary point, while "<" would exclude it.

Solving inequalities often uses number lines, graphing, or algebraic manipulation. Understanding inequalities in graphical settings aid in visualizing potential solutions and comprehending the interaction of differing mathematical expressions.