Find composite function expression

5 The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 3\) for \(x \geqslant 0\).

1. Find and simplify an expression for \(\mathrm { ff } ( x )\).
2. Solve the equation \(\mathrm { ff } ( x ) = 34 x ^ { 2 } + 19\).

11 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } f ( x ) = x ^ { 2 } + 3 & \text { for } x > 0 \\ g ( x ) = 2 x + 1 & \text { for } x > - \frac { 1 } { 2 } \end{array}$$

1. Find an expression for \(\mathrm { fg } ( x )\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
3. Solve the equation \(\mathrm { fg } ( x ) - 3 = \mathrm { gf } ( x )\).

11 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 \\ & \mathrm {~g} : x \mapsto x ^ { 2 } - 2 \end{aligned}$$

1. Find and simplify expressions for \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Hence find the value of \(a\) for which \(\mathrm { fg } ( a ) = \mathrm { gf } ( a )\).
3. Find the value of \(b ( b \neq a )\) for which \(\mathrm { g } ( b ) = b\).
4. Find and simplify an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\). The function h is defined by $$\mathrm { h } : x \mapsto x ^ { 2 } - 2 , \quad \text { for } x \leqslant 0$$
5. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

10 The function f is such that \(\mathrm { f } ( x ) = 2 x + 3\) for \(x \geqslant 0\). The function g is such that \(\mathrm { g } ( x ) = a x ^ { 2 } + b\) for \(x \leqslant q\), where \(a , b\) and \(q\) are constants. The function fg is such that \(\operatorname { fg } ( x ) = 6 x ^ { 2 } - 21\) for \(x \leqslant q\).

1. Find the values of \(a\) and \(b\).
2. Find the greatest possible value of \(q\). It is now given that \(q = - 3\).
3. Find the range of fg.
4. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).

8 The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x ^ { 2 } + 3 \\ & \mathrm {~g} : x \mapsto 3 x + 2 \end{aligned}$$

1. Show that \(\operatorname { gf } ( x ) = 6 x ^ { 2 } + 11\) and obtain an unsimplified expression for \(\operatorname { fg } ( x )\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and determine the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
3. Solve the equation \(\mathrm { gf } ( 2 x ) = \mathrm { fg } ( x )\).

10 The function f is defined by $$\mathrm { f } : x \mapsto 3 x - 2 \text { for } x \in \mathbb { R } .$$

1. 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. The function g is defined by $$\mathrm { g } : x \mapsto 6 x - x ^ { 2 } \text { for } x \in \mathbb { R }$$
2. Express \(\operatorname { gf } ( x )\) in terms of \(x\), and hence show that the maximum value of \(\operatorname { gf } ( x )\) is 9 . The function h is defined by $$\mathrm { h } : x \mapsto 6 x - x ^ { 2 } \text { for } x \geqslant 3$$
3. Express \(6 x - x ^ { 2 }\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants.
4. Express \(\mathrm { h } ^ { - 1 } ( x )\) in terms of \(x\).

3 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 3 \\ & \mathrm {~g} : x \mapsto x ^ { 2 } - 2 x \end{aligned}$$ Express \(\operatorname { gf } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

2 The functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + a \\ & \mathrm {~g} : x \mapsto b - 2 x \end{aligned}$$ where \(a\) and \(b\) are constants. Given that \(\mathrm { ff } ( 2 ) = 10\) and \(\mathrm { g } ^ { - 1 } ( 2 ) = 3\), find

1. the values of \(a\) and \(b\),
2. an expression for \(\mathrm { fg } ( x )\).

8 The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 4 } { x } - 2 \quad \text { for } x > 0 \\ & \mathrm {~g} ( x ) = \frac { 4 } { 5 x + 2 } \quad \text { for } x \geqslant 0 \end{aligned}$$

1. Find and simplify an expression for \(\mathrm { fg } ( x )\) and state the range of fg.
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { g } ^ { - 1 }\).

9 Functions f and g are defined for \(x > 3\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { x ^ { 2 } - 9 } \\ & \mathrm {~g} : x \mapsto 2 x - 3 \end{aligned}$$

1. Find and simplify an expression for \(\operatorname { gg } ( x )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Solve the equation \(\operatorname { fg } ( x ) = \frac { 1 } { 7 }\).

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 3 } { 2 x + 1 } \quad \text { for } x > 0 \\ & \mathrm {~g} : x \mapsto \frac { 1 } { x } + 2 \quad \text { for } x > 0 \end{aligned}$$

1. Find the range of f and the range of g .
2. Find an expression for \(\mathrm { fg } ( x )\), giving your answer in the form \(\frac { a x } { b x + c }\), where \(a , b\) and \(c\) are integers.
3. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\), giving your answer in the same form as for part (ii).

1. Given that

$$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 } { 3 x + 5 } , & x > 0 \\ \mathrm {~g} ( x ) = \frac { 1 } { x } , & x > 0 \end{array}$$

1. state the range of f,
2. find \(\mathrm { f } ^ { - 1 } ( x )\),
3. find \(\mathrm { fg } ( x )\).
4. Show that the equation \(\mathrm { fg } ( x ) = \mathrm { gf } ( x )\) has no real solutions.

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$ The graph cuts the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 2 .

1. State the range of g .
2. State the coordinates of
   1. point \(A\)
   2. point \(B\)
3. Find \(\operatorname { gg } ( x )\) in its simplest form.
4. Sketch the graph with equation \(y = | \mathrm { g } ( x ) |\) On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
5. Find the exact solution of the equation \(| \mathrm { g } ( x ) | = 8\)

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \mapsto 1 - 2 x ^ { 3 } , x \in \mathbb { R } \\ & \mathrm {~g} : x \mapsto \frac { 3 } { x } - 4 , x > 0 , x \in \mathbb { R } \end{aligned}$$

1. Find the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Show that the composite function gf is $$\text { gf } : x \mapsto \frac { 8 x ^ { 3 } - 1 } { 1 - 2 x ^ { 3 } }$$
3. Solve \(\operatorname { gf } ( x ) = 0\).
4. Use calculus to find the coordinates of the stationary point on the graph of \(y = \operatorname { gf } ( x )\).

2. The function \(f\) is defined by $$\mathrm { f } : x \mapsto 2 x , \quad x \in \mathbb { R }$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } : x \mapsto 3 x ^ { 2 } + 2 , \quad x \in \mathbb { R }$$
2. Find \(\mathrm { gf } ^ { - 1 } ( x )\).
3. State the range of \(\mathrm { gf } ^ { - 1 } ( x )\).

2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for all real numbers \(x\) by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = x - 2$$

1. Find the composite functions \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Sketch the curves \(y = \mathrm { f } ( x ) , y = \mathrm { fg } ( x )\) and \(y = \mathrm { gf } ( x )\), indicating clearly which is which.

3 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for the domain \(x > 0\) as follows: $$\mathrm { f } ( x ) = \ln x , \quad \mathrm {~g} ( x ) = x ^ { 3 } .$$ Express the composite function \(\mathrm { fg } ( x )\) in terms of \(\ln x\).
State the transformation which maps the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { fg } ( x )\).

2 Given that \(\mathrm { f } ( x ) = 1 - x\) and \(\mathrm { g } ( x ) = | x |\), write down the composite function \(\mathrm { gf } ( x )\).
On separate diagrams, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { gf } ( x )\).

7 The functions \(f , g\) and \(h\) are defined as follows. $$\mathrm { f } ( x ) = 2 x \quad \mathrm {~g} ( x ) = x ^ { 2 } \quad \mathrm {~h} ( x ) = x + 2$$ Find each of the following as functions of \(x\).

1. \(\mathrm { f } ^ { 2 } ( x )\),
2. \(\operatorname { fgh } ( x )\),
3. \(\mathrm { h } ^ { - 1 } ( x )\).

2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for all real numbers \(x\) by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = x - 2 .$$

1. Find the composite functions \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Sketch the curves \(y = \mathrm { f } ( x ) , y = \mathrm { fg } ( x )\) and \(y = \mathrm { gf } ( x )\), indicating clearly which is which.

3 Given that \(\mathrm { f } ( x ) = 1 - x\) and \(\mathrm { g } ( x ) = | x |\), write down the composite function \(\mathrm { gf } ( x )\). On separate diagrams, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { gf } ( x )\).

4 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = \sqrt [ 3 ] { x - 10 }$$

1. Evaluate \(\mathrm { f } ^ { - 1 } ( - 50 )\).
2. Show that \(\operatorname { fg } ( x ) = 2 x - 16\).
3. Differentiate \(\operatorname { gf } ( x )\) with respect to \(x\).

4. Given $$\begin{aligned} & \mathrm { f } ( x ) = \mathrm { e } ^ { x } , \quad x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 3 \ln x , \quad x > 0 , x \in \mathbb { R } \end{aligned}$$

1. find an expression for \(\mathrm { gf } ( x )\), simplifying your answer.
2. Show that there is only one real value of \(x\) for which \(\operatorname { gf } ( x ) = \operatorname { fg } ( x )\)

9 You are given that \(\mathrm { f } ( x ) = 2 x + 3 \quad\) for \(x < 0 \quad\) and \(\mathrm { g } ( x ) = x ^ { 2 } - 2 x + 1\) for \(x > 1\).

1. Find \(\mathrm { gf } ( x )\), stating the domain.
2. State the range of \(\mathrm { gf } ( x )\).
3. Find (gf) \({ } ^ { - 1 } ( x )\).

15 Functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined as follows. \(\mathrm { f } ( x ) = \sqrt { x }\) for \(x > 0\) and \(\mathrm { g } ( x ) = x ^ { 3 } - x - 6\) for \(x > 2\). The function \(\mathrm { h } ( x )\) is defined as \(\mathrm { h } ( x ) = \mathrm { fg } ( x )\).

1. Find \(\mathrm { h } ( x )\) in terms of \(x\) and state its domain.
2. Find \(\mathrm { h } ( 3 )\). Fig. 15 shows \(\mathrm { h } ( x )\) and \(\mathrm { h } ^ { - 1 } ( x )\), together with the straight line \(y = x\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-17_780_796_895_242} \captionsetup{labelformat=empty} \caption{Fig. 15}
   \end{figure}
3. Determine the gradient of \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) at the point where \(y = 3\).