The function g is defined by g(x)=2x^2-12x+13 for x>=4.
Expalin why g has an inverse.
Review you definition of an inverse.
What are the steps in finding the inverse that you have learned?
Hint.
g(x) = 2(x-3)^2 - 5
The vertex is at x=3
To explain why function g has an inverse, first, we need to understand what an inverse function is. An inverse function undoes the actions performed by the original function, resulting in the original input value.
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).