Range of f : R → Y

To find the range of the function f(x) = 5x^2 - 6x - 9, we need to determine the set of all possible y-values that can be obtained by plugging in different x-values into the function.

Finding the Range

To find the range, we can start by considering the leading term of the function, which is 5x^2. Since the coefficient of x^2 is positive, the parabola opens upwards. This means that the function has a minimum value, and all y-values greater than or equal to this minimum value can be obtained.

Finding the Vertex

To find the minimum value and the x-coordinate of the vertex, we can use the formula x = -b/2a. In this case, a = 5 and b = -6.

x = -(-6) / (2 * 5) = 6/10 = 3/5

To find the corresponding y-coordinate of the vertex, we can substitute the x-coordinate into the function:

f(3/5) = 5(3/5)^2 - 6(3/5) - 9 = 5(9/25) - 18/5 - 9 = 45/25 - 90/25 - 225/25 = -270/25 = -54/5

So, the vertex of the parabola is (3/5, -54/5).

Range

Since the parabola opens upwards and the vertex is the minimum point, the range of the function is all y-values greater than or equal to the y-coordinate of the vertex.

Range of f: y ≥ -54/5

Invertibility of f

To determine if the function f(x) = 5x^2 - 6x - 9 is invertible, we need to check if it satisfies the necessary conditions for invertibility.

One-to-One Function

A function is one-to-one if every distinct x-value corresponds to a distinct y-value, and vice versa. In other words, no two different x-values can produce the same y-value.

To check if f(x) is one-to-one, we can use the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

Horizontal Line Test

The graph of the function f(x) = 5x^2 - 6x - 9 is a parabola that opens upwards. Since the vertex is the minimum point, the graph is symmetric with respect to the vertical line passing through the vertex.

Therefore, no horizontal line intersects the graph at more than one point. Hence, the function f(x) is one-to-one.

Existence of Inverse Function

Since f(x) is one-to-one, it implies that an inverse function exists. The inverse function is denoted as f^(-1)(y) and is defined as the function that undoes the actions of f(x).

To find the inverse function, we can follow these steps:

1. Replace f(x) with y: y = 5x^2 - 6x - 9
2. Swap x and y: x =