Math  /  Algebra

QuestionThe sketch below represents the graphs of f(x)=2x31f(x)=\frac{2}{x-3}-1 and g(x)=dx+eg(x)=d x+e. Point B(3;6)\mathrm{B}(3 ; 6) lies on the graph of gg and the two graphs intersect at A and C . 7.1 Write down the equations of the asymptotes of ff. 7.2 Write down the domain of ff. 7.3 Determine the values of dd and ee, correct to the nearest integer, if the graph of gg makes an angle of 7676^{\circ} with the xx axis. 7.4 Determine the coordinates of A and C . 7.5 For which value(s) of xx is: 7.5.1 f(x)g(x)f(x) \leq g(x) ? 7.5.2 x. f(x)0f(x) \geq 0 ? [20]

Studdy Solution
Solve the inequality f(x)g(x) f(x) \leq g(x) by finding the intervals where this condition holds.
Solve the inequality f(x)0 f(x) \geq 0 to find the values of x x .
The equations of the asymptotes of f(x) f(x) are: Vertical: x=3 x = 3 Horizontal: y=1 y = -1
The domain of f(x) f(x) is: xR,x3 x \in \mathbb{R}, x \neq 3
The values of d d and e e are: d=round(tan(76)) d = \text{round}(\tan(76^\circ)) e=63d e = 6 - 3d
The coordinates of A and C are determined by solving the intersection equations.
The values of x x for which f(x)g(x) f(x) \leq g(x) and f(x)0 f(x) \geq 0 are found through solving the inequalities.

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