Page 2: Properties and Domain of Exponential Functions

This page delves into the properties of exponential functions and explores various scenarios for determining their domains.

Vocabulary: The dominio funzione esponenziale $domain of an exponential function$ refers to the set of all possible input values for which the function is defined.

The page lists key properties of exponents, such as:

• a^x * a^y = a^$x+y$
• $a^x$^y = a^$xy$

It then presents several examples of exponential functions and their domains, including:

• f$x$ = 2^$x-2$
• f$x$ = 5^$√(x+2$)
• f$x$ = $√3$^$(x^2+1$/$x^2-4$)

Example: For f$x$ = 5^$√(x+2$), the domain is x ≥ -2, as the expression under the square root must be non-negative.

The page emphasizes the importance of considering the base and exponent when determining the domain of an exponential function.

Page 3: Solving Exponential Equations

This page focuses on techniques for solving exponential equations, which are crucial for understanding potenze con esponente reale esercizi svolti $solved exercises with real exponents$.

Definition: An exponential equation is an equation where the unknown appears in the exponent.

The page outlines two main strategies for solving exponential equations:

1. For equations of the form a^f$x$ = a^g$x$, set f$x$ = g$x$.
2. For equations of the form a^x = b^x, transform both sides to have the same base.

Example: To solve 3^$x+2$ = 9^$x-1$, rewrite as 3^$x+2$ = $3^2$^$x-1$, then equate exponents: x + 2 = 2$x-1$.

The page provides several worked examples, demonstrating how to apply these techniques to solve complex exponential equations.

Page 4: Advanced Exponential Equations

This page continues with more complex exponential equations, building on the techniques introduced earlier.

Highlight: When solving exponential equations, it's often helpful to use logarithms or to make substitutions to simplify the problem.

The page presents several challenging examples, including:

• 2^$x^2 - 5x$ = 2^x
• 3^$2x+1$ + 3^x = 40

For each example, the page provides a step-by-step solution, demonstrating various problem-solving strategies such as:

• Equating exponents
• Using the change of variable technique
• Applying quadratic equation solving methods

Example: To solve 3^$2x+1$ + 3^x = 40, let y = 3^x. This transforms the equation to 3y + y = 40, which can be solved as a quadratic equation in y.

These examples help students develop problem-solving skills for tackling complex potenze con esponente reale Zanichelli exercises.

Page 5: Exponential Inequalities

This page introduces exponential inequalities, an important topic in disequazioni esponenziali $exponential inequalities$.

Definition: An exponential inequality is an inequality that involves exponential expressions.

The page emphasizes two key points when solving exponential inequalities:

1. For exponential functions with base a > 1, the inequality sign remains the same.
2. For exponential functions with base 0 < a < 1, the inequality sign reverses.

Example: For 2^x < 8, since 2 > 1, we can take logarithms without changing the inequality: log₂$2^x$ < log₂8, which simplifies to x < 3.

The page provides several examples of exponential inequalities and their solutions, including:

• 3^$x-2$ < 3^$-2x-1$
• $1/2$^$2x+2$ > $1/3$^$x-2$

It also covers more complex cases involving compound inequalities and change of variable techniques.

Page 6: Exponential Growth

This page explores the concept of exponential growth, a key application of exponential functions in real-world scenarios.

Definition: Exponential growth occurs when the rate of change of a quantity is proportional to its current value.

The page uses the example of bacterial growth to illustrate exponential growth:

• Starting with 10 bacteria
• Population increases by 120% each hour

Example: The bacterial population after t hours can be modeled by the function m$t$ = 10$2.2$^t.

The page demonstrates how to use this model to predict the bacterial population after 8 hours, resulting in approximately 5,488 bacteria.

Highlight: Exponential growth models are widely used in various fields, including biology, finance, and physics, to make predictions and analyze rapidly changing phenomena.

Page 7: Complex Exponential Functions

This final page examines more complex exponential functions, particularly those of the form y = $f(x)$^g$x$.

Vocabulary: The dominio funzioni esponenziali $domain of exponential functions$ becomes more complex when dealing with composite functions.

The page analyzes three main cases:

1. y = a^f$x$, where a > 0
2. y = $f(x)$^a, where a is a real number
3. y = $f(x)$^g$x$

For each case, the page provides examples and explains how to determine the domain of the function.

Example: For y = $x^2 - 5$^√$x^2 - 1$, the domain is determined by two conditions:

1. x^2 - 1 ≥ 0 $for the square root in the exponent$
2. x^2 - 5 > 0 $for the base to be positive$

The page concludes by emphasizing the importance of considering both the base and exponent when determining the domain of complex exponential functions.

Page 1: Introduction to Exponential Functions

Exponential functions are defined as y = a^x, where a is a positive real number not equal to 1. This page introduces the basic concepts and characteristics of these functions.

Definition: An exponential function is of the form y = a^x, where a > 0 and a ≠ 1. The number a is called the base of the exponential function.

The page distinguishes between three cases:

1. 0 < a < 1: Decreasing function
2. a = 1: Constant function $excluded from the definition$
3. a > 1: Increasing function

Highlight: All exponential functions are strictly positive, with their graphs lying above the x-axis in the first and second quadrants.

The page also notes that all exponential functions pass through the point $0, 1$ and that functions with reciprocal bases $e.g., 2^x and (1/2$^x) have graphs that are symmetric about the y-axis.