Problem 60 - College Algebra Essentials

Question

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. Graph \(y=3^{x}\) and \(x=3^{y}\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The functions \(f(x) = 3^{x}\) and \(g(y) = 3^{y}\) intersect at the point (1,1)

Step 1: Plot Function 1: \(f(x)=3^{x}\)

Plot the function \(f(x) = 3^{x}\) on a graph. Draw an x-y axis and then plot the function by substituting various x-values to get the corresponding y-values.

Step 2: Plot Function 2: \(g(y) = 3^{y}\)

On the same graph, plot the function \(g(y) = 3^{y}\) by substituting various y values and plotting the resulting x-values.

Step 3: Find Intersection Point

The intersection point will be a point that lies on both lines. Look for the point or points where the lines intersect, which represent the x and y values that satisfy both functions. In this particular case, the two functions \(y=3^{x}\) and \(x=3^{y}\) intersect at the point (1,1).

Key Concepts

Understanding the Coordinate System

The coordinate system is fundamental to graphing functions. Most commonly, we use a rectangular or Cartesian coordinate system. This system consists of two perpendicular axes: the horizontal axis known as the x-axis, and the vertical axis called the y-axis. These axes intersect at the origin, which is denoted by the point (0,0). The coordinate plane is divided into four quadrants, labeled from I to IV starting from the top right quadrant and moving counterclockwise.

• Quadrant I: both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: both x and y are negative.
• Quadrant IV: x is positive, y is negative.

When graphing functions like exponential ones, it's important to correctly place the values based on their x and y coordinates. For any point on the graph, the x-value represents the horizontal position, while the y-value indicates the vertical position. Accurately plotting points on this grid helps in visualizing the behavior of functions such as intersections of graphs.

Exploring Exponential Functions

Exponential functions are critical in mathematics due to their unique properties. A basic exponential function can be expressed generally as \(f(x) = a^{x}\), where \(a\) is a constant greater than zero, and not equal to one. These functions are characterized by the rate at which they grow - or decay - being proportional to their current value.

• If \(a > 1\), the function represents exponential growth.
• If \(0 < a < 1\), it represents exponential decay.

In the problem at hand, we look at \(f(x) = 3^{x}\), an example of exponential growth. As x increases, \(3^x\) rapidly increases, creating a steep upward curve on the graph. These functions never touch the x-axis, but approach it indefinitely as x becomes more negative. Understanding this behavior is vital when examining and plotting these types of functions, especially when analyzing intersections with other curves.

Techniques for Plotting Graphs

Plotting graphs involves systematically placing points on a coordinate system and connecting them to visualize a function. This step makes abstract math concepts tangible and easier to understand. To plot a function, start by choosing a set of x-values and calculate the corresponding y-values using the function's equation. For example, with \(f(x) = 3^x\), you can choose x-values like -2, -1, 0, 1, and 2, and find their related y-values.

• For \(x = 0\), \(f(0) = 3^0 = 1\).
• For \(x = 1\), \(f(1) = 3^1 = 3\).
• For \(x = 2\), \(f(2) = 3^2 = 9\).

And similarly for other x-values. Plot these points carefully on the coordinate system and connect them smoothly to form the curve.
Repeat a similar process for \(g(y) = 3^{y}\), with your possible y-values. When both graphs are plotted, look for where they intersect; those points represent solutions satisfying both equations. Identifying the intersection visually integrates algebraic and graphical understanding, confirming that (1,1) satisfies both functions in our specific example.