Problem 63 - Precalculus with Calculus Previews

Question

Graph the given functions. Determine the approximate \(x\) -coordinates of the points of intersection of their graphs. $$ f(x)=3^{x}, g(x)=2\left(3^{x}\right) $$

Step-by-Step Solution

Answer

The functions do not intersect, as they are identical in shape but differ by a constant factor.

Step 1: Understand the Functions

The given functions are \( f(x) = 3^x \) and \( g(x) = 2(3^x) \). Notice that \( g(x) \) is simply twice the value of \( f(x) \), suggesting that \( g(x) \) is just a vertical stretch of \( f(x) \).

Step 2: Graph the Functions

To graph the functions, start with \( f(x) = 3^x \), which is an exponential curve starting from \( (0, 1) \), rising to the right since \( 3^x \) increases as \( x \) increases. Then, \( g(x) = 2 \times 3^x \) should also be plotted on the same set of axes. It will follow the same shape but should be above \( f(x) \) because of the factor of 2.

Step 3: Analyze Intersection Points

Determine the intersections by setting \( f(x) = g(x) \), which gives \( 3^x = 2(3^x) \). Simplify to find \( 3^x - 2(3^x) = 0 \), then \( -3^x = 0 \). This equation shows there are no real values of \( x \) for which the graphs intersect because \( -3^x \) will not equal zero.

Key Concepts

Understanding Exponential Functions

Exponential functions are a special type of mathematical function where the variable appears in the exponent. In general, an exponential function can be expressed as \( a^x \), where \( a \) is a constant known as the base and \( x \) is the exponent. One key feature of exponential functions is their rapid rate of growth or decay, depending on whether the base is greater or less than one.
In our example, we have two functions: \( f(x) = 3^x \) and \( g(x) = 2(3^x) \). Both functions showcase how exponential growth behaves with a base of 3, meaning for every increase in \( x \), the value of \( f(x) \) or \( g(x) \) increases at an accelerating rate. This nature of exponential functions makes them ideal for modeling scenarios involving growth processes, such as population growth or compound interest.

Graphing Functions For Visual Understanding

Graphing functions is a useful technique for visualizing their behavior. When graphing \( f(x) = 3^x \), start by selecting a few key points. At \( x = 0 \), \( 3^x \) equals 1, giving the point (0, 1). As \( x \) increases, \( 3^x \) increases, resulting in a curve that rises steeply as it moves from left to right. Plotting several points will help to draw this exponential curve.
The second function, \( g(x) = 2(3^x) \), keeps the same basic structure as \( f(x) \), since it's just a scaled version. This function will start at (0, 2) because \( g(0) = 2(1) \). Plotting similar points as you did for \( f(x) \), you'll notice \( g(x) \) runs parallel and above \( f(x) \), representing the doubling of \( f(x) \) values. To determine potential intersections, normally you would set \( f(x) = g(x) \). However, as demonstrated, these graphs do not intersect, because the equation \( 3^x = 2(3^x) \) simplifies in a way that shows there are no real solutions.

The Concept of Vertical Stretch

A vertical stretch in functions occurs when every output value (\( y \)-value) of a function is multiplied by a constant. This alters the steepness of the graph without changing its general shape or location horizontally. In math terms, if you take a function \( f(x) \) and create a new function \( kf(x) \), where \( k \) is a constant greater than 1, you've created a vertical stretch.
In our example, \( g(x) = 2(3^x) \) represents a vertical stretch of \( f(x) = 3^x \) by a factor of 2. This means that at every point \( x \), the output of \( g(x) \) is double that of \( f(x) \). Importantly, while the height of the graph changes, the exponential nature and base stay the same. Vertical stretching doesn't affect the x-coordinates of points on the graph, just their height, making the concept a simple yet powerful tool in transforming functions.