Problem 58 - Algebra and Trigonometry

Question

graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. $$ f(x)=2^{x+1}, g(x)=2^{-x+1} $$

Step-by-Step Solution

Answer

The points of intersection of the functions \(f(x)=2^{x+1}\) and \(g(x)=2^{-x+1}\) are at (0,2) and (-2/3, 4/3).

Step 1: Plot the Functions

To plot the functions \(f(x)=2^{x+1}\) and \(g(x)=2^{-x+1}\), several points for x and corresponding f(x) or g(x) should be picked out and plotted on a graph. For this problem, values of x can be chosen from -3 to 3 to see a clear indication of how the functions behave.

Step 2: Find Points of Intersection

To find the points of intersection, we need to set equations of the functions equal to each other and solve for x. \n So, \(2^{x+1} = 2^{-x+1}\). To make the bases the same and simplify the equation further, apply the property \(a^{-n} = \frac{1}{a^n}\). Thus, the equation becomes \(2^{x+1} = \frac{1}{2^{x-1}}\). Cross multiplying gives \(2^{2x+1} = 2^{x-1}\). Equating the exponents gives two solutions, \(2x+1 = x-1\) and \(2x+1= -(x-1)\) which result in \(x = 0\) and \(x = -2/3\). Substituting these values into either of the original function equations provides the corresponding y-values, yielding points of intersection.

Step 3: Verify the Solution

To verify the accuracy of the solutions, these points can then be plotted on the graph of the functions. If the points lie on both graphs, the solutions are correct.

Key Concepts

Graphing Functions

Graphing functions involves visually representing each function on a coordinate plane. It helps us understand how functions behave and how they're related to each other. To begin with, when you have equations like \(f(x)=2^{x+1}\) and \(g(x)=2^{-x+1}\), you first determine the values of the function by substituting different x-values. Once the pairs \((x,f(x))\) and \((x,g(x))\) are determined, these points are plotted on the graph.

For instance, choosing values of \(x\) from -3 to 3 gives a complete picture of how the functions behave. The importance of graphing is that it gives you a visual insight into the solution of equations, making it easier to predict where the functions might intersect.

Coordinate System

A coordinate system allows us to visually represent points on a plane using horizontal and vertical lines, known as axes. The most common is the rectangular coordinate system, also known as the Cartesian plane. This consists of the x-axis (horizontal) and y-axis (vertical) intersecting at a point called the origin, noted as \((0, 0)\).

When plotting functions like \(f(x)=2^{x+1}\) and \(g(x)=2^{-x+1}\), each point \((x, y)\) on these graphs uses the coordinate system to show where it falls on the plane. Understanding the coordinate system is crucial because it forms the foundation of graphing functions. The axes act as a reference that helps you determine where points lie relative to each other.

Exponential Functions

Exponential functions are a critical concept in mathematics, characterized by variables in the exponent. For example, in \(f(x)=2^{x+1}\), the variable \(x\) is in the exponent, making this an exponential function. This property causes exponential functions to exhibit rapid changes, often modeling growth or decay.

In exponential functions, the base greater than 1 generally indicates growth (as in \(f(x)=2^{x+1}\)), while a base of less than 1 (when written in terms of negative exponents, like \(g(x)=2^{-x+1}\)) indicates decay.

• Rapid increase or decrease in value for small changes in x.
• Constant ratio of change, unlike linear functions.

Understanding these characteristics helps in predicting how the function will behave and in solving equations that involve exponential dynamics.

Solving Equations

Solving equations involves finding values for the variable that make the equation true. In the given exercise, you had to solve \(2^{x+1} = 2^{-x+1}\). The solution process involves understanding the properties of exponents and simplifying the equation to find common solutions for both sides of the equation.

The steps include:

• Rewriting one side to match the format of the other side, if necessary.
• Equating the exponents when bases are the same. This is done by setting \(2x+1 = x-1\) and solving, and \(2x+1 = -(x-1)\).
• Solving these simpler linear equations gives you the \(x\)-values where the original functions intersect.

Once \(x\) is found, you substitute back into one of the function equations to get the corresponding \(y\)-values. Double-checking the points against the graph ensures they accurately reflect the intersection points.