Question

In Exercises 31-34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range. $$ y=2 e^x+1 $$

Short answer

The domain of the function \(y = 2e^x + 1\) is all real numbers, or (-∞, ∞), and the range is all real numbers greater than 1, or (1, ∞).

Step-by-step solution

Graphing the function

To graph the function \(y=2e^x+1\), either make a table of values using different x-values and their corresponding y-values, or use a graphing calculator. It should be noted that this function indicates an exponential curve.

Identifying the Domain

In this function, x could be any real number. Hence, the domain is all real numbers. In interval notation, this is written as (-∞, ∞)

Identifying the Range

For this function, because of the nature of the exponential function, y-values would always be greater than 1. Hence, the range is written in interval notation as (1, ∞).

Key concepts

Graphing Calculators

Graphing calculators are powerful tools that help you visualize functions quickly and easily. For students working with exponential functions like \( y = 2e^x + 1 \), these calculators can swiftly generate a graph to illustrate the growth that the equation describes. By inputting the function into the graphing calculator, you can obtain a visual depiction which makes it simpler to understand how the function behaves.

Here’s what a graphing calculator does when you input an exponential function:

• It plots many points, providing a smooth curve.
• Shows the exponential growth pattern clearly.
• Helps identify key features like intersections, if any.

Enjoying a graphing calculator's ease can foster deeper comprehension, as seeing the curve can help in understanding the abstract mathematical concept represented by the formula.

Domain and Range

Understanding the domain and range of a function is crucial in math, especially for exponential functions. Let's delve deeper into these concepts using the function \( y = 2e^x + 1 \):

- **Domain**: This encompasses all the possible x-values that can be input into the function. For our exponential function, because there's no restriction on x, the domain is all real numbers, from negative infinity to positive infinity. In mathematical terms, this can be conveyed as \((-\infty, \infty)\). Thus, you can input any real number into \( y = 2e^x + 1 \) and obtain a valid output.
- **Range**: The range is the set of possible output values (y-values) that the function can provide. In this exponential function, we'll notice that as you plug in any x-value, the resulting y-value will always be greater than 1. This is because the smallest value of \( e^x \) is 0 when \( x \to -\infty \), meaning \( y = 2e^x + 1 \) always yields results larger than 1. Hence, the range for this function is \((1, \infty)\). Knowing this helps give clarity on what the function can produce.

Exponential Growth

Exponential growth is a crucial concept when dealing with exponential functions like \( y = 2e^x + 1 \). This type of growth is characterized by the rate of change being proportional to the current value, leading to faster increases as the values become larger.

Here's what happens in an exponential growth scenario:

• The base of the exponent (in this case, e from \( e^x \)) is greater than 1, contributing to the continuous increase.
• The graph features an upward curve that becomes steeper as x becomes larger.
• Small increases in x result in larger increases in \( y \) as the x-value gets bigger, illustrating the rapid growth rate.

This rapid growth is seen in various fields like finance, biology, and physics, making it an essential concept to grasp. Understanding exponential growth through functions like \( y = 2e^x + 1 \) equips students with the knowledge to tackle real-world applications efficiently.