The features of the graph of the function f(x) = (-x+4)/(2x-8) are: domain of all real numbers except x = 4, range of all real numbers, vertical asymptote at x = 4, horizontal asymptote at y = -1/2, x-intercept at (4, 0), y-intercept at (0, -1/2), no symmetry, no critical points, no local extrema, and decreasing on its entire domain.
To determine the features of the graph of the function f(x) = (-x+4)/(2x-8), we can analyze its properties and use calculus.
Domain and Range: The function is defined for all x except x = 4, which makes the denominator zero. Therefore, the domain of the function is all real numbers except x = 4. The range of the function is all real numbers.
Vertical Asymptote: The function has a vertical asymptote at x = 4, where the denominator is zero.
Horizontal Asymptote: To find the horizontal asymptote, we can use long division or synthetic division to divide the numerator by the denominator. The result is:
(-x+4)/(2x-8) = (-1/2) + (5/4)/(x-4)
As x approaches infinity or negative infinity, the fraction (5/4)/(x-4) approaches zero, so the horizontal asymptote is y = -1/2.
Intercepts: To find the x-intercept, we set y = 0 and solve for x:
(-x+4)/(2x-8) = 0
-x + 4 = 0
x = 4
Therefore, the x-intercept is (4, 0).
To find the y-intercept, we set x = 0 and solve for y:
(-0+4)/(2(0)-8) = -1/2
Therefore, the y-intercept is (0, -1/2).
Symmetry: The function is not symmetric with respect to the x-axis, y-axis, or origin.
Derivative and Critical Points: To find the derivative of the function, we can use the quotient rule:
f'(x) = [(2x-8)(-1) - (-x+4)(2)] / (2x-8)^2
Simplifying, we get:
f'(x) = -6 / (2x-8)^2
The derivative is never zero, so there are no critical points.
Increasing and Decreasing Intervals: Since the derivative is always negative, the function is decreasing on its entire domain.
Local Extrema: Since the function is decreasing on its entire domain, there are no local extrema.
Therefore, the features of the graph of the function f(x) = (-x+4)/(2x-8) are: domain of all real numbers except x = 4, range of all real numbers, vertical asymptote at x = 4, horizontal asymptote at y = -1/2, x-intercept at (4, 0), y-intercept at (0, -1/2), no symmetry, no critical points, no local extrema, and decreasing on its entire domain.