Simplify then show identity

1. The function f is defined by

$$f ( x ) = \frac { 2 x ^ { 2 } - 32 } { 3 x ^ { 2 } + 7 x - 20 } + \frac { 8 } { 3 x - 5 } \quad x \in \mathbb { R } \quad x > 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 5 }\)
2. Show, using calculus, that f is a decreasing function. You must make your reasoning clear. The function g is defined by $$g ( x ) = 3 + 2 \ln x \quad x \geqslant 1$$
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 5$$

4. $$\mathrm { f } ( x ) = x + \frac { 3 } { x - 1 } - \frac { 12 } { x ^ { 2 } + 2 x - 3 } , x \in \mathbb { R } , x > 1$$

1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 3 } { x + 3 }\).
2. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = \frac { 22 } { 25 }\).

1. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 }$$ Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$

1. The function \(f\) is given by

$$\mathrm { f } : x \propto \frac { x } { x ^ { 2 } - 1 } - \frac { 1 } { x + 1 } , x > 1$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { ( x - 1 ) ( x + 1 ) }\).
2. Find the range of f. The function \(g\) is given by $$\mathrm { g } : x \propto \frac { 2 } { x } , x > 0$$
3. Solve \(\operatorname { gf } ( x ) = 70\).

11 In this question you must show detailed reasoning. A function f is given by \(\mathrm { f } ( x ) = \frac { x - 4 } { ( x + 2 ) ( x - 1 ) } + \frac { 3 x + 1 } { ( x + 3 ) ( x - 1 ) }\).

1. Show that \(\mathrm { f } ( x )\) can be written as \(\frac { 2 ( 2 x + 5 ) } { ( x + 2 ) ( x + 3 ) }\).
2. Given that \(\int _ { a } ^ { a + 4 } \mathrm { f } ( x ) \mathrm { d } x = 2 \ln 3\), find the value of the positive constant \(a\).