Find composite function expression

3 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 - x ^ { 2 } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } , & \text { for real values of } x , x \neq - 1 \end{array}$$

1. Find the range of f.
2. The inverse of g is \(\mathrm { g } ^ { - 1 }\).
   1. Find \(\mathrm { g } ^ { - 1 } ( x )\).
   2. State the range of \(\mathrm { g } ^ { - 1 }\).
3. The composite function gf is denoted by h .
   1. Find \(\mathrm { h } ( x )\), simplifying your answer.
   2. State the greatest possible domain of h .

8 The curve with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }\), is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}

1. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. State the range of \(\mathrm { f } ^ { - 1 }\).
   3. Sketch, on the axes given on page 9 , the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating the value of the \(y\)-coordinate of the point where the curve intersects the \(y\)-axis.
      (2 marks)
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
   1. Find \(\mathrm { gf } ( x )\), giving your answer in the form \(( a x - b ) ^ { 2 } - c\), where \(a , b\) and \(c\) are integers.
   2. Write down an expression for \(\mathrm { fg } ( x )\), and hence find the exact solution of the equation \(\operatorname { fg } ( x ) = \ln 5\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_1079_1422_233_358}

5 The function f is defined by $$\mathrm { f } ( x ) = 16 x - \mathrm { e } ^ { 2 x } , \text { for all real } x$$ The graph of \(y = \mathrm { f } ( x )\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-12_789_1349_534_347}

1. Find the range of f.
2. The composite function fg is defined by $$\operatorname { fg } ( x ) = \frac { 16 } { x } - \mathrm { e } ^ { \frac { 2 } { x } } , \text { for real } x , x \neq 0$$ Find an expression for \(\operatorname { gg } ( x )\).

8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , \quad x \neq - 2 \end{array}$$

1. State the range of f.
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
3. 1. Explain why the function f does not have an inverse.
   2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).

8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , x \neq - 2 \end{array}$$

1. State the range of f.
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
3. 1. Explain why the function f does not have an inverse.
   2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).

5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 - x ^ { 4 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x - 4 } & \text { for real values of } x , x \neq 4 \end{array}$$

1. State the range of f .
2. Explain why the function f does not have an inverse.
3. 1. Write down an expression for fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = - 14\).

5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { x - 2 } \text { for } x \geqslant 2 \\ & \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for real values of } x , x \neq 0 \end{aligned}$$

1. State the range of f .
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 1\).
3. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\).

11 For all real values of \(x\), the functions f and g are defined by \(\mathrm { f } ( x ) = x ^ { 2 } + 8 a x + 4 a ^ { 2 }\) and \(g ( x ) = 6 x - 2 a\), where \(a\) is a positive constant.

1. Find \(\mathrm { fg } ( x )\). Determine the range of \(\mathrm { fg } ( x )\) in terms of \(a\).
2. If \(f g ( 2 ) = 144\), find the value of \(a\).
3. Determine whether the function fg has an inverse.

6 The functions f and g are given by \(\mathrm { f } ( x ) = \frac { 3 } { x - 1 }\) for all \(x \neq 1\) and \(\mathrm { g } ( x ) = x + 2\) for all real \(x\).

1. Find gf, stating its domain and range.
2. Find \(( \mathrm { gf } ) ^ { - 1 }\), stating any values of \(x\) for which \(( \mathrm { gf } ) ^ { - 1 }\) is not defined.

The function f is defined by \(\text{f}(x) = \frac{48}{x - 1}\) for \(3 \leqslant x \leqslant 7\). The function g is defined by \(\text{g}(x) = 2x - 4\) for \(a \leqslant x \leqslant b\), where \(a\) and \(b\) are constants.

1. Find the greatest value of \(a\) and the least value of \(b\) which will permit the formation of the composite function gf. [2] It is now given that the conditions for the formation of gf are satisfied.
2. Find an expression for \(\text{gf}(x)\). [1]
3. Find an expression for \((\text{gf})^{-1}(x)\). [2]

$$f(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find the range of \(f\). [1]
2. Write down the domain and range of \(f^{-1}\). [2]
3. Sketch the graph of \(f^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]

Given that \(g(x) = |x - 4|, x \in \mathbb{R}\),

1. find an expression for \(gf(x)\). [2]
2. Solve \(gf(x) = 8\). [5]

f(x) = \(x^2 - 2x - 3\), \(x \in \mathbb{R}\), \(x \geq 1\).

1. Find the range of f. [1]
2. Write down the domain and range of \(f^{-1}\). [2]
3. Sketch the graph of \(f^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]

Given that g(x) = \(|x - 4|\), \(x \in \mathbb{R}\),

1. find an expression for gf(x). [2]
2. Solve gf(x) = 8. [5]

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

The functions \(\text{f}(x)\) and \(\text{g}(x)\) are defined as follows. $$\text{f}(x) = \ln x, \quad x > 0$$ $$\text{g}(x) = 1 + x^2, \quad x \in \mathbb{R}$$ Write down the functions \(\text{fg}(x)\) and \(\text{gf}(x)\), and state whether these functions are odd, even or neither. [4]

Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function \(gf(x)\), expressing your answer as simply as possible. [3]

The functions f(x) and g(x) are defined as follows. $$\text{f}(x) = \ln x, \quad x > 0$$ $$\text{g}(x) = 1 + x^2, \quad x \in \mathbb{R}$$ Write down the functions fg(x) and gf(x), and state whether these functions are odd, even or neither. [4]

Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function gf(x), expressing your answer as simply as possible. [3]

$$f(x) = e^x, x \in \mathbb{R}, x > 0.$$ $$g(x) = 2x^3 + 11, x \in \mathbb{R}.$$

1. Find and simplify an expression for the composite function \(gf(x)\). [2]
2. State the domain and range of \(gf(x)\). [2]
3. Solve the equation $$gf(x) = 27.$$ [3]

The equation \(gf(x) = k\), where \(k\) is a constant, has solutions.

1. State the range of the possible values of \(k\). [1]

In this question you must show detailed reasoning. The functions f and g are defined for all real values of \(x\) by $$f(x) = x^3 \text{ and } g(x) = x^2 + 2.$$

1. Write down expressions for
   1. \(fg(x)\), [1]
   2. \(gf(x)\). [1]
2. Hence find the values of \(x\) for which \(fg(x) - gf(x) = 24\). [6]

For all real values of \(x\), the functions \(f\) and \(g\) are defined by \(f (x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.

1. Find \(fg(x)\). Determine the range of \(fg(x)\) in terms of \(a\). [4]
2. If \(fg(2) = 144\), find the value of \(a\). [3]
3. Determine whether the function \(fg\) has an inverse. [2]

For all real values of \(x\), the functions f and g are defined by \(f(x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.

1. Find fg\((x)\). Determine the range of fg\((x)\) in terms of \(a\). [4]
2. If fg\((2) = 144\), find the value of \(a\). [3]
3. Determine whether the function fg has an inverse. [2]

% Figure 4 shows curves with asymptotic behavior at x = 3 \includegraphics{figure_4} Figure 4

1. Figure 4 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 - 5}{3-x}, \quad x \in \mathbb{R}, x \neq 3$$ The curve has a minimum at the point \(A\), with \(x\)-coordinate \(\alpha\), and a maximum at the point \(B\), with \(x\)-coordinate \(\beta\). Find the value of \(\alpha\), the value of \(\beta\) and the \(y\)-coordinates of the points \(A\) and \(B\). [5]
2. The functions \(g\) and \(h\) are defined as follows $$g: x \to x + p \quad x \in \mathbb{R}$$ $$h: x \to |x| \quad x \in \mathbb{R}$$ where \(p\) is a constant. % Figure 5 shows curve with minimum points at C and D symmetric about y-axis \includegraphics{figure_5} Figure 5 Figure 5 shows a sketch of the curve with equation \(y = h(fg(x) + q)\), \(x \in \mathbb{R}\), \(x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).
   1. Find the value of \(p\) and the value of \(q\).
   2. Write down the coordinates of \(D\).
   [5]
3. The function \(\mathrm{m}\) is given by $$\mathrm{m}(x) = \frac{x^2 - 5}{3-x} \quad x \in \mathbb{R}, x < \alpha$$ where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
   1. Find \(\mathrm{m}^{-1}\)
   2. Write down the domain of \(\mathrm{m}^{-1}\)
   3. Find the value of \(t\) such that \(\mathrm{m}(t) = \mathrm{m}^{-1}(t)\)
   [10]

[Total 20 marks]