given f(x)=-2/x-2 +2 determine the equation of the vertical asymptote ' ' ' ' horizontal asymptote 2.find the coordinates of x intercept. 3.' ' ' y ' '. 4.sketch the graph of of f. 5.Determine the equations of the axes of symmetry of f(x)

To determine the equation of the vertical asymptote of the function f(x), you need to find the x-value(s) at which the function becomes undefined. In this case, the function is undefined when the denominator of a fraction becomes zero because division by zero is undefined.

In the given function, the denominator is x - 2. To find the vertical asymptote, set the denominator equal to zero and solve for x:

x - 2 = 0
x = 2

Therefore, the equation of the vertical asymptote is x = 2.

To find the horizontal asymptote, you need to determine the behavior of the function as x approaches positive or negative infinity. To do this, compare the powers of x in the numerator and denominator.

In the given function, the highest power of x is 1 in both the numerator and denominator. Therefore, you divide the coefficient of the highest power term in the numerator by the coefficient of the highest power term in the denominator:

-2/1 = -2

The value -2 represents the horizontal asymptote of the function f(x).

To find the x-intercepts, you need to determine the points at which the function crosses the x-axis. To do this, set f(x) equal to 0 and solve for x:

-2 / (x - 2) + 2 = 0

First, subtract 2 from both sides:

-2 / (x - 2) = -2

Now, cross multiply:

-2 = -2(x - 2) = -2x + 4

Simplify:

-2 + 4 = -2x
2 = -2x

Divide by -2:

x = -1

Therefore, the x-intercept of the function is (-1, 0).

To find the y-intercept, substitute x = 0 into the function:

f(0) = -2 / (0 - 2) + 2
f(0) = -2 / (-2) + 2
f(0) = 1 + 2
f(0) = 3

Therefore, the y-intercept of the function is (0, 3).

To sketch the graph of the function f(x), you can use the information about the vertical asymptote at x = 2, the horizontal asymptote at y = -2, the x-intercept at (-1, 0), and the y-intercept at (0, 3). Additionally, you can plot a few more points by substituting different x-values into the function and calculating the corresponding y-values.

Finally, to determine the equation of the axis of symmetry, you need to find the line that passes through the vertex of the graph, which is the point where the function reaches its maximum or minimum. However, in this case, the function does not have a vertex because it is a rational function.

Therefore, the equation of the axis of symmetry is not applicable for this function.