Review you definition of an inverse.
What are the steps in finding the inverse that you have learned?
Expalin why g has an inverse.
What are the steps in finding the inverse that you have learned?
g(x) = 2(x-3)^2 - 5
The vertex is at x=3
In order for a function to have an inverse, it must be a one-to-one function. A one-to-one function means that each input value corresponds to a unique output value, and each output value has a unique input value.
To determine if function g is one-to-one, we can use a method called the horizontal line test. It involves checking if any horizontal line intersects the graph of function g more than once.
The function g(x) = 2x^2 - 12x + 13 is a quadratic function, and the graph of a quadratic function is a parabola. Since the coefficient of x^2 is positive, the parabola opens upwards. Therefore, the vertex of the parabola represents the lowest point, and any horizontal line will intersect the graph at most once.
Hence, the graph of function g passes the horizontal line test, indicating that it is a one-to-one function. Therefore, g has an inverse.
If we want to find the inverse function, we can follow these steps:
Step 1: Replace g(x) with y
y = 2x^2 - 12x + 13
Step 2: Swap x and y variables
x = 2y^2 - 12y + 13
Step 3: Solve for y
Rearrange the equation:
2y^2 - 12y + (13 - x) = 0
Apply the quadratic formula:
y = (12 ± √(12^2 - 4 * 2 * (13 - x))) / (2 * 2)
Simplify:
y = (6 ± √(36 - 8(13 - x))) / 4
y = (6 ± √(36 - 104 + 8x)) / 4
y = (6 ± √(8x - 68)) / 4
y = (3 ± √(2x - 17)) / 2
The resulting expression, (3 ± √(2x - 17)) / 2, represents the inverse function of g(x).