Solve equation with inverses

5 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 4 x - 2 , \quad \text { for } x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) = \frac { 4 } { x + 1 } , \quad \text { for } x \in \mathbb { R } , x \neq - 1 \end{aligned}$$

1. Find the value of fg (7).
2. Find the values of \(x\) for which \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).

2 Functions \(f\) and \(g\) are defined by $$\begin{aligned} \mathrm { f } : x & \mapsto 3 x + 2 , \quad x \in \mathbb { R } , \\ \mathrm {~g} : x & \mapsto 4 x - 12 , \quad x \in \mathbb { R } . \end{aligned}$$ Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { gf } ( x )\).

10 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 3 x + 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \frac { 6 } { 2 x + 3 } , & x \in \mathbb { R } , x \neq - 1.5 . \end{array}$$

1. Find the value of \(x\) for which \(\operatorname { fg } ( x ) = 3\).
2. Sketch, in a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the two graphs.
3. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\), and solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto k - x & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, } \\ \mathrm { g } : x \mapsto \frac { 9 } { x + 2 } & \text { for } x \in \mathbb { R } , x \neq - 2 . \end{array}$$

1. Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has two equal roots and solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) in these cases.
2. Solve the equation \(\operatorname { fg } ( x ) = 5\) when \(k = 6\).
3. Express \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).

9 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 2 } { 3 - 2 x }\) for \(x \in \mathbb { R } , x \neq \frac { 3 } { 2 }\).

1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } : x \mapsto 4 x + a\) for \(x \in \mathbb { R }\), where \(a\) is a constant.
2. Find the value of \(a\) for which \(\operatorname { gf } ( - 1 ) = 3\).
3. Find the possible values of \(a\) given that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has two equal roots.

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 2 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 2 x + 3 } { x - 1 } , \quad x \in \mathbb { R } , x \neq 1 \end{aligned}$$

1. Obtain expressions for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined. [4]
2. Solve the equation \(\operatorname { fg } ( x ) = \frac { 7 } { 3 }\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 5 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 4 } { 2 - x } , \quad x \in \mathbb { R } , \quad x \neq 2 . \end{aligned}$$

1. Find the value of \(x\) for which \(\mathrm { fg } ( x ) = 7\).
2. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
3. Show that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no real roots.
4. Sketch, on a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between these two graphs.

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 , \quad x \in \mathbb { R } , \quad x > 0 \\ & \mathrm {~g} : x \mapsto \frac { 2 x - 1 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$

1. Solve the equation \(\operatorname { gf } ( x ) = x\).
2. Express \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
3. Show that the equation \(\mathrm { g } ^ { - 1 } ( x ) = x\) has no solutions.
4. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.

1 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + 2 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto 4 x - 12 , \quad x \in \mathbb { R } . \end{aligned}$$ Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \operatorname { gf } ( x )\).

8. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$

1. Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\) and give its domain.
2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\). The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
3. Find the value of \(k\).

6. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$

1. State the range of f.
2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3 .$$
4. Evaluate \(\mathrm { fg } ( - 3 )\).
5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x ) .$$

Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \begin{align} \mathrm{f} : x \mapsto 2x + 3 \quad &\text{for } x \leqslant 0,
\mathrm{g} : x \mapsto x^2 - 6x \quad &\text{for } x \leqslant 3. \end{align}

1. Express \(\mathrm{f}^{-1}(x)\) in terms of \(x\) and solve the equation \(\mathrm{f}(x) = \mathrm{f}^{-1}(x)\). [3]
2. On the same diagram sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\), showing the coordinates of their point of intersection and the relationship between the graphs. [3]
3. Find the set of values of \(x\) which satisfy \(\mathrm{gf}(x) \leqslant 16\). [5]

The functions f and g are defined for all real values of \(x\) by $$\text{f}(x) = x^2 + 4ax + a^2 \text{ and } \text{g}(x) = 4x - 2a,$$ where \(a\) is a positive constant.

1. Find the range of f in terms of \(a\). [4]
2. Given that fg(3) = 69, find the value of \(a\) and hence find the value of \(x\) such that \(\text{g}^{-1}(x) = x\). [6]

The function f is defined by $$f(x) = 2x + 1$$ Solve the equation $$f(x) = f^{-1}(x)$$ Circle your answer. [1 mark] \(x = -1\) \quad\quad \(x = 0\) \quad\quad \(x = 1\) \quad\quad \(x = 2\)