based on the function f(x)= 2x-4/x+4 list the following characteristics?

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

x -intercept when f(x) = 0, i.e. when 2x - 4 = 0

=> x = 2...i.e. point (2, 0)

y-intercept when x = 0, so at (0, -1)

The function is not defined when x = -4, hence domain x ≠ -4

i.e. the line x = -4 is a vertical asymptote

Now, f(x) => 2(x + 4)/(x + 4) - 12/(x + 4)

i.e. 2 - 12/(x + 4)

Also, f '(x) = 12/(x + 4)²

Now, for any value of x, 12/(x + 4)² > 0...i.e. always positive

Hence, in the interval (-∞, -4) and (-4, ∞) the gradient is increasing.

Considering f(x) => 2 - 12/(x + 4) we can see that as x --> -∞, -12/(x + 4) --> 0 with ever decreasing positive values. Conversely, as x --> +∞, -12/(x + 4) --> 0 with ever decreasing negative values.

Hence, as f(x) --> 2....horizontal asymptote

For, x < -4, f(x) --> 2 from above the line f(x) = 2

For, x > -4, f(x) --> 2 from below the line f(x) = 2

f(x) is positive when x < -4 and when x > 2

f(x) is negative when -4 < x < 2

The sketch is below.

https://www.desmos.com/calculator/xroj2smb7l

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