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.