Evaluate composite at point

6

1. The function f , defined by \(\mathrm { f } : x \mapsto a + b \sin x\) for \(x \in \mathbb { R }\), is such that \(\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3\).
   1. Find the values of the constants \(a\) and \(b\).
   2. Evaluate \(\mathrm { ff } ( 0 )\).
2. The function g is defined by \(\mathrm { g } : x \mapsto c + d \sin x\) for \(x \in \mathbb { R }\). The range of g is given by \(- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10\). Find the values of the constants \(c\) and \(d\).

2. The function \(f\) and the function \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 12 } { x + 1 } & x > 0 , x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \frac { 5 } { 2 } \ln x & x > 0 , x \in \mathbb { R } \end{array}$$

1. Find, in simplest form, the value of \(\mathrm { fg } \left( \mathrm { e } ^ { 2 } \right)\)
2. Find f-1
3. Hence, or otherwise, find all real solutions of the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )$$

1. The functions f and g are defined by

$$\begin{array} { l l l } \mathrm { f } ( x ) = 9 - x ^ { 2 } & x \in \mathbb { R } & x \geqslant 0 \\ \mathrm {~g} ( x ) = \frac { 3 } { 2 x + 1 } & x \in \mathbb { R } & x \geqslant 0 \end{array}$$

1. Write down the range of f
2. Find the value of fg(1.5)
3. Find \(\mathrm { g } ^ { - 1 }\)

2. The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 5 - x } { 3 x + 2 } & x \in \mathbb { R } , x \neq - \frac { 2 } { 3 } \\ \mathrm {~g} ( x ) = 2 x - 7 & x \in \mathbb { R } \end{array}$$

1. Find the value of \(\mathrm { fg } ( 5 )\)
2. Find \(\mathrm { f } ^ { - 1 }\)
3. Solve the equation $$\mathrm { f } \left( \frac { 1 } { a } \right) = \mathrm { g } ( a + 3 )$$

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { x + 3 } { x - 4 } \quad x \in \mathbb { R } , x \neq 4$$

1. Find ff(6)
2. Find \(f ^ { - 1 }\) The function \(g\) is defined by $$g ( x ) = x ^ { 2 } + 5 \quad x \in \mathbb { R } , x > 0$$
3. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 7$$

7. For the constant \(k\), where \(k > 1\), the functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \ln ( x + k ) , \quad x > - k , \\ & \mathrm {~g} : x \mapsto | 2 x - k | , \quad x \in \mathbb { R } . \end{aligned}$$

1. On separate axes, sketch the graph of f and the graph of g . On each sketch state, in terms of \(k\), the coordinates of points where the graph meets the coordinate axes.
2. Write down the range of f.
3. Find \(\mathrm { fg } \left( \frac { k } { 4 } \right)\) in terms of \(k\), giving your answer in its simplest form. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The tangent to \(C\) at the point with \(x\)-coordinate 3 is parallel to the line with equation \(9 y = 2 x + 1\).
4. Find the value of \(k\).

1. The functions f and g are defined by

$$\begin{array} { l l } \mathrm { f } : x \mapsto 2 | x | + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto 3 - 4 x , & x \in \mathbb { R } \end{array}$$

1. State the range of f.
2. Find \(\mathrm { fg } ( 1 )\).
3. Find \(\mathrm { g } ^ { - 1 }\), the inverse function of g .
4. Solve the equation $$\operatorname { gg } ( x ) + [ \mathrm { g } ( x ) ] ^ { 2 } = 0$$

7. The function \(f\) has domain \(- 2 \leqslant x \leqslant 6\) and is linear from \(( - 2,10 )\) to \(( 2,0 )\) and from \(( 2,0 )\) to (6, 4). A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h] \end{figure}

1. Write down the range of f .
2. Find \(\mathrm { ff } ( 0 )\). The function \(g\) is defined by $$\mathrm { g } : x \rightarrow \frac { 4 + 3 x } { 5 - x } , \quad x \in \mathbb { R } , \quad x \neq 5$$
3. Find \(\mathrm { g } ^ { - 1 } ( x )\)
4. Solve the equation \(\operatorname { gf } ( x ) = 16\)

6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 3 x - 4 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \frac { 2 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$

1. Evaluate fg(1).
2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

4 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_529_737_900_701} The function f is defined by \(\mathrm { f } ( x ) = 2 - \sqrt { x }\) for \(x \geqslant 0\). The graph of \(y = \mathrm { f } ( x )\) is shown above.

1. State the range of f.
2. Find the value of \(\mathrm { ff } ( 4 )\).
3. Given that the equation \(| \mathrm { f } ( x ) | = k\) has two distinct roots, determine the possible values of the constant \(k\).

1 Functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = 2 x - 5$$ Evaluate

1. \(f g ( 1 )\),
2. \(\mathrm { f } ^ { - 1 } ( 12 )\).

1 The function f is defined for all real values of \(x\) by $$f ( x ) = 10 - ( x + 3 ) ^ { 2 } .$$

1. State the range of f .
2. Find the value of \(\mathrm { ff } ( - 1 )\).

5 \includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-3_844_837_242_621} It is given that f is a one-one function defined for all real values. The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). The coordinates of certain points on the curve are shown in the following table.

1. State the value of \(\mathrm { ff } ( 6 )\) and the value of \(\mathrm { f } ^ { - 1 } ( 8 )\).
2. On the copy of the diagram, sketch the curve \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating how the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related.
3. Use Simpson's rule with 6 strips to find an approximation to \(\int _ { 2 } ^ { 14 } \mathrm { f } ( x ) \mathrm { d } x\).

10. The functions f and g are defined with their respective domains by $$\begin{array} { l l l } \mathrm { f } ( x ) = 4 - x ^ { 2 } & x \in R & x \geq 0 \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } & x \in R & x \geq 0 \end{array}$$ a. Write down the range of f .
b. Find the value of \(\mathrm { fg } ( 3 )\) c. Find \(\mathrm { g } ^ { - 1 } ( x )\)

1. The function f is defined by

$$f ( x ) = 3 + \sqrt { x - 2 } \quad x \in \mathbb { R } \quad x > 2$$

1. State the range of f
2. Find f-1 The function \(g\) is defined by $$g ( x ) = \frac { 15 } { x - 3 } \quad x \in \mathbb { R } \quad x \neq 3$$
3. Find \(g f ( 6 )\)
4. Find the exact value of the constant \(a\) for which $$\mathrm { f } \left( a ^ { 2 } + 2 \right) = \mathrm { g } ( a )$$

1. The functions f and g are defined by

$$\begin{aligned} & f ( x ) = 7 - 2 x ^ { 2 } \quad x \in \mathbb { R } \\ & \operatorname { g } ( x ) = \frac { 3 x } { 5 x - 1 } \quad x \in \mathbb { R } \quad x \neq \frac { 1 } { 5 } \end{aligned}$$

1. State the range of f
2. Find gf (1.8)
3. Find \(\mathrm { g } ^ { - 1 } ( x )\)

4 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a , \\ \ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$ The range of f is \(\mathrm { f } ( x ) \geqslant - 2\).

1. Show that \(a = \frac { 3 } { 2 }\).
2. Find the value of \(b\).
3. Find the exact value of \(\mathrm { ff } ( - 1 )\).
4. Explain why the function f does not have an inverse.

1. The functions f and g are defined by

$$\begin{array} { l l } \mathrm { f } ( x ) = 6 - \frac { 21 } { 2 x + 3 } & x \geqslant 0 \\ \mathrm {~g} ( x ) = x ^ { 2 } + 5 & x \in \mathbb { R } \end{array}$$

1. Find \(\mathrm { gf } ( 2 )\)
2. Find \(f ^ { - 1 }\)
3. Solve the equation $$\operatorname { gg } ( x ) = 126$$

The functions \(f\) and \(g\) are defined by $$f: x \mapsto x^2 - 2x + 3, x \in \mathbb{R}, \quad 0 \leq x \leq 4,$$ $$g: x \mapsto \lambda x^2 + 1, \text{ where } \lambda \text{ is a constant, } x \in \mathbb{R}.$$

1. Find the range of \(f\). [3]
2. Given that \(gf(2) = 16\), find the value of \(\lambda\). [3]

The functions f and g are defined by \(\text{f: } x \mapsto x^2 - 2x + 3, x \in \mathbb{R}, 0 \leq x \leq 4,\) \(\text{g: } x \mapsto \lambda x^2 + 1, \text{ where } \lambda \text{ is a constant, } x \in \mathbb{R}.\)

1. Find the range of f. [3]
2. Given that gf(2) = 16, find the value of \(\lambda\). [3]

The function f is defined for all real values of \(x\) by $$f(x) = 10 - (x + 3)^2.$$

1. State the range of f. [1]
2. Find the value of ff(-1). [3]

Functions f and g are defined for all real values of \(x\) by $$f(x) = x^3 + 4 \quad \text{and} \quad g(x) = 2x - 5.$$ Evaluate

1. fg(1), [2]
2. \(f^{-1}(12)\). [3]