To find the values of x for which f(x) = g(x), we need to set the two functions equal to each other and solve for x.
Setting f(x) = g(x), we have:
x^2 - 1 = 3^x
This equation is not easily solved algebraically, and its solution would involve using numerical methods or approximation techniques. However, we can analyze the behavior of the two functions to gain some insights.
The function f(x) = x^2 - 1 is a quadratic function that opens upwards and has a vertex at (0, -1). It increases as x moves away from the vertex, forming a U-shaped curve.
The function g(x) = 3^x is an exponential function with a base of 3. It increases rapidly as x becomes larger and decreases as x becomes smaller. The graph of g(x) is an upward-sloping curve that grows exponentially.
By observing the behavior of the two functions, we can conclude that there will be at most two points of intersection, if any. These points of intersection represent the values of x for which f(x) and g(x) are equal.
To find the exact values of x, we can use numerical methods such as graphing the functions and finding the intersection points, or using iterative methods like the bisection method or Newton's method.
Therefore, the values of x for which f(x) = g(x) can be determined using numerical methods.