Problem 21 - Precalculus with Calculus Previews

Question

Determine the range of the given function. $$ f(x)=3^{x}-2 $$

Step-by-Step Solution

Answer

The range of the function \( f(x) = 3^x - 2 \) is \((-2, \, \infty)\).

Step 1: Identify the base function

The function given is of the form \( f(x) = 3^x - 2 \). The base function here is \( g(x) = 3^x \), which is an exponential function. The range of \( g(x) = 3^x \) is all positive real numbers \((0, \, \infty)\).

Step 2: Effect of transformation

The given function \( f(x) = 3^x - 2 \) represents a vertical shift of the base function \( g(x) = 3^x \) downward by 2 units. This vertical shift affects the range of the function.

Step 3: Determine new range

Consider the lowest value that \( 3^x \) can approach, which in this case is slightly above 0. With the downward shift by 2, the lowest value \( 3^x \) approaches (0) will transform to -2. The new range of \( f(x) \) becomes slightly above -2 to infinity, which is \((-2, \, \infty)\).

Key Concepts

Understanding Exponential Functions

Exponential functions are a fundamental part of mathematics, characterized by a constant base raised to a variable exponent. A common form is \( g(x) = a^x \), where \( a \) is a positive real number greater than 1. In the given exercise, our base function is \( g(x) = 3^x \). This specific type of function is known for its rapid growth, as it can increase much faster than linear or polynomial functions.

• Key Characteristics: One core feature of exponential functions is that they are always positive, meaning they never intersect the x-axis and hover entirely above it.
• They also possess a horizontal asymptote at \( y=0 \), indicating that as \( x \) approaches negative infinity, the function values get arbitrarily close to 0 but never become zero.

Knowing the behavior of exponential functions helps us comprehend how transformations like vertical shifts affect their range.

Vertical Shift and Its Impact

A vertical shift in a function occurs when a constant is added or subtracted from the function's output, moving its graph up or down along the y-axis. For the function \( f(x) = 3^x - 2 \), subtracting 2 results in a downward vertical shift.

• The result is that all the values of the function \( 3^x \) move two units lower.
• This operation doesn't affect the domain but crucially alters the range.

Due to this vertical shift, the horizontal asymptote of \( g(x) = 3^x \) at \( y=0 \) shifts to \( y=-2 \). This change is vital for determining the function's new range.
By understanding vertical shifts, you can predict how they transform the appearance and properties of a function, which is especially useful when determining behavior around asymptotes.

Transformation Effect on the Function's Range

Transformations alter functions in various ways. In our problem, the transformation is a vertical shift of an exponential function, leading us to a new range.

• The original exponential function \( g(x) = 3^x \) ranges over all positive numbers, \( (0, \, \infty) \).
• After applying the vertical shift, the transformed function \( f(x) = 3^x - 2 \) has a range of \( (-2, \, \infty) \).

This is because the function values above zero are shifted down by two, thus the new lowest bound of the range becomes just above -2.
When finding the range, always consider how each transformation like shifting, scaling, or reflecting reshapes the original function. Understanding these effects will make it easier to predict and sketch these functions.