Graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. f(x) = 2^x, g(x) = 2^-x

Verified step by step guidance

1. First, understand the functions given: \(f(x) = 2^x\) is an exponential growth function, and \(g(x) = 2^{-x}\) is an exponential decay function. Both are defined for all real numbers \(x\).
2. To graph both functions on the same coordinate system, create a table of values for each function by choosing several \(x\) values (for example, \(-2, -1, 0, 1, 2\)) and calculating the corresponding \(f(x)\) and \(g(x)\) values.
3. Plot the points from the tables for \(f(x)\) and \(g(x)\) on the coordinate plane. Remember that \(f(x) = 2^x\) increases as \(x\) increases, while \(g(x) = 2^{-x}\) decreases as \(x\) increases.
4. To find the point of intersection algebraically, set the two functions equal to each other: \(2^x = 2^{-x}\). This equation will help find the \(x\)-value(s) where the graphs intersect.
5. Solve the equation \(2^x = 2^{-x}\) by using properties of exponents. For example, rewrite \(2^{-x}\) as \(\frac{1}{2^x}\) and solve for \(x\). Once you find \(x\), substitute it back into either \(f(x)\) or \(g(x)\) to find the corresponding \(y\)-coordinate of the intersection point.

Key Concepts

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They model rapid growth or decay depending on whether the exponent is positive or negative. Understanding their shape and behavior is essential for graphing and analyzing functions like f(x) = 2^x and g(x) = 2^-x.

Graphing Functions on the Coordinate Plane

Graphing involves plotting points (x, f(x)) on the rectangular coordinate system to visualize the function's behavior. Comparing two graphs on the same axes helps identify intersections and relative positions. Accurate plotting of exponential functions reveals their growth and decay patterns.

Finding Points of Intersection

The point of intersection of two graphs is where their function values are equal, i.e., f(x) = g(x). Solving this equation algebraically or by inspection gives the x-coordinate(s) of intersection. Substituting back finds the corresponding y-coordinate(s), providing the exact intersection point(s).