1.02v Inverse and composite functions: graphs and conditions for existence

\includegraphics{figure_8} The diagram shows the graph of \(y = f(x)\) where the function \(f\) is defined by $$f(x) = 3 + 2\sin \frac{1}{4}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$

1. On the diagram above, sketch the graph of \(y = f^{-1}(x)\). [2]
2. Find an expression for \(f^{-1}(x)\). [2]
3. \includegraphics{figure_8c} The diagram above shows part of the graph of the function \(g(x) = 3 + 2\sin \frac{1}{4}x\) for \(-2\pi \leqslant x \leqslant 2\pi\). Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]
4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\) to the graph of \(y = f(x)\), making clear the order in which the transformations are applied. [6]

\includegraphics{figure_6} The function f is defined by f\((x) = \frac{2}{x^2} + 4\) for \(x < 0\). The diagram shows the graph of \(y = \text{f}(x)\).

1. On this diagram, sketch the graph of \(y = \text{f}^{-1}(x)\). Show any relevant mirror line. [2]
2. Find an expression for f\(^{-1}(x)\). [3]
3. Solve the equation f\((x) = 4.5\). [1]
4. Explain why the equation f\(^{-1}(x) = \text{f}(x)\) has no solution. [1]

The function f is defined as follows: $$f(x) = \sqrt{x-1} \text{ for } x > 1.$$ \begin{enumerate}[label=(\alph*)] \item Find an expression for \(f^{-1}(x)\). [1] \end enumerate} \includegraphics{figure_4} The diagram shows the graph of \(y = g(x)\) where \(g(x) = \frac{1}{x^2+2}\) for \(x \in \mathbb{R}\). \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item State the range of g and explain whether \(g^{-1}\) exists. [2] \end enumerate} The function h is defined by \(h(x) = \frac{1}{x^2+2}\) for \(x \geqslant 0\). \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{2} \item Solve the equation \(hf(x) = f\left(\frac{25}{16}\right)\). Give your answer in the form \(a + b\sqrt{c}\), where \(a\), \(b\) and \(c\) are integers. [4] \end enumerate}

Functions f and g are defined by $$f(x) = (x + a)^2 - a \text{ for } x \leqslant -a,$$ $$g(x) = 2x - 1 \text{ for } x \in \mathbb{R},$$ where \(a\) is a positive constant.

1. Find an expression for \(f^{-1}(x)\). [3]
2. 1. State the domain of the function \(f^{-1}\). [1]
   2. State the range of the function \(f^{-1}\). [1]
3. Given that \(a = \frac{7}{2}\), solve the equation \(gf(x) = 0\). [3]

The function f is defined by f\((x) = 3 + 6x - 2x^2\) for \(x \in \mathbb{R}\).

1. Express f\((x)\) in the form \(a - b(x - c)^2\), where \(a\), \(b\) and \(c\) are constants, and state the range of f. [3]
2. The graph of \(y = \)f\((x)\) is transformed to the graph of \(y = \)h\((x)\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \)h\((x)\) has a minimum point at the origin. Give details of the reflection and translation involved. [2] The function g is defined by g\((x) = 3 + 6x - 2x^2\) for \(x \leqslant 0\).
3. Sketch the graph of \(y = \)g\((x)\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]
4. Sketch the graph of \(y = \)g\(^{-1}(x)\) on your diagram in (c), and find an expression for g\(^{-1}(x)\). You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]

The function f is defined by \(\mathrm{f}(x) = \frac{2x + 1}{2x - 1}\) for \(x < \frac{1}{2}\).

1. 1. State the value of f\((-1)\). [1]
   2. \includegraphics{figure_5} The diagram shows the graph of \(y = \mathrm{f}(x)\). Sketch the graph of \(y = \mathrm{f}^{-1}(x)\) on this diagram. Show any relevant mirror line. [2]
   3. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of the function \(\mathrm{f}^{-1}\). [4]

The function g is defined by \(\mathrm{g}(x) = 3x + 2\) for \(x \in \mathbb{R}\).

1. Solve the equation \(\mathrm{f}(x) = \mathrm{gf}\left(\frac{1}{4}\right)\). [3]

1. Express \(3x^2 - 12x + 14\) in the form \(3(x + a)^2 + b\), where \(a\) and \(b\) are constants to be found. [2]

The function f(x) = \(3x^2 - 12x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.

1. Find the least value of \(k\) for which the function \(\text{f}^{-1}\) exists. [1]

For the rest of this question, you should assume that \(k\) has the value found in part (b).

1. Find an expression for \(\text{f}^{-1}(x)\). [3]
2. Hence or otherwise solve the equation \(\text{f f}(x) = 29\). [3]

The functions f and g are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto 4x - 2x^2,$$ $$g : x \mapsto 5x + 3.$$

1. Find the range of f. [2]
2. Find the value of the constant \(k\) for which the equation \(gf(x) = k\) has equal roots. [3]

The function \(f : x \mapsto 4 - 3\sin x\) is defined for the domain \(0 \leq x < 2\pi\).

1. Solve the equation \(f(x) = 2\). [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Find the set of values of \(k\) for which the equation \(f(x) = k\) has no solution. [2]

The function \(g : x \mapsto 4 - 3\sin x\) is defined for the domain \(\frac{1}{2}\pi \leq x \leq A\).

1. State the largest value of \(A\) for which \(g\) has an inverse. [1]
2. For this value of \(A\), find the value of \(g^{-1}(3)\). [2]

The function \(f\) is defined by \(f : x \mapsto \frac{x + 3}{2x - 1}\), \(x \in \mathbb{R}\), \(x \neq \frac{1}{2}\).

1. Show that \(f f(x) = x\). [3]
2. Hence, or otherwise, obtain an expression for \(f^{-1}(x)\). [2]

The function \(f\) is such that \(f(x) = 3 - 4\cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.

1. In the case where \(k = 2\),
   1. find the range of \(f\), [2]
   2. find the exact solutions of the equation \(f(x) = 1\). [3]
2. In the case where \(k = 1\),
   1. sketch the graph of \(y = f(x)\), [2]
   2. state, with a reason, whether \(f\) has an inverse. [1]

Functions \(f\) and \(g\) are defined by $$f : x \mapsto 2x + 5 \quad \text{for } x \in \mathbb{R},$$ $$g : x \mapsto \frac{8}{x - 3} \quad \text{for } x \in \mathbb{R}, x \neq 3.$$

1. Obtain expressions, in terms of \(x\), for \(f^{-1}(x)\) and \(g^{-1}(x)\), stating the value of \(x\) for which \(g^{-1}(x)\) is not defined. [4]
2. Sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) on the same diagram, making clear the relationship between the two graphs. [3]
3. Given that the equation \(fg(x) = 5 - kx\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\). [5]

The function \(f\) is such that \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\).

1. Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\). [3]

The function \(g\) is such that \(g(x) = 8 - (x - 2)^2\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which \(g\) has an inverse. [1]

For this value of \(k\),

1. find an expression for \(g^{-1}(x)\), [3]
2. sketch, on the same diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\). [3]

The function \(\text{f} : x \mapsto 5 + 3\cos(\frac{1}{3}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Solve the equation \(\text{f}(x) = 7\), giving your answer correct to 2 decimal places. [3]
2. Sketch the graph of \(y = \text{f}(x)\). [2]
3. Explain why \(\text{f}\) has an inverse. [1]
4. Obtain an expression for \(\text{f}^{-1}(x)\). [3]

The function f is defined by \(\mathrm{f} : x \mapsto 2x^2 - 6x + 5\) for \(x \in \mathbb{R}\).

1. Find the set of values of \(p\) for which the equation \(\mathrm{f}(x) = p\) has no real roots. [3]

The function g is defined by \(\mathrm{g} : x \mapsto 2x^2 - 6x + 5\) for \(0 \leqslant x \leqslant 4\).

1. Express \(\mathrm{g}(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Find the range of g. [2]

The function h is defined by \(\mathrm{h} : x \mapsto 2x^2 - 6x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which h has an inverse. [1]
2. For this value of \(k\), find an expression for \(\mathrm{h}^{-1}(x)\). [3]

1. Express \(9x^2 - 6x + 6\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]

The function f is defined by \(\text{f}(x) = 9x^2 - 6x + 6\) for \(x \geqslant p\), where \(p\) is a constant.

1. State the smallest value of \(p\) for which f is a one-one function. [1]
2. For this value of \(p\), obtain an expression for \(\text{f}^{-1}(x)\), and state the domain of \(\text{f}^{-1}\). [4]
3. State the set of values of \(q\) for which the equation \(\text{f}(x) = q\) has no solution. [1]

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]

1. Express \(x^2 - 4x + 7\) in the form \((x + a)^2 + b\). [2]

The function \(f\) is defined by \(f(x) = x^2 - 4x + 7\) for \(x < k\), where \(k\) is a constant.

1. State the largest value of \(k\) for which \(f\) is a decreasing function. [1]

The value of \(k\) is now given to be \(1\).

1. Find an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\). [3]
2. The function \(g\) is defined by \(g(x) = \frac{2}{x-1}\) for \(x > 1\). Find an expression for \(gf(x)\) and state the range of \(gf\). [4]

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 function \(f : x \mapsto 6 - 4\cos(\frac{1}{2}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Find the exact value of \(x\) for which \(f(x) = 4\). [3]
2. State the range of \(f\). [2]
3. Sketch the graph of \(y = f(x)\). [2]
4. Find an expression for \(f^{-1}(x)\). [3]

Functions f and g are defined by $$f : x \mapsto 2 - 3\cos x \text{ for } 0 \leqslant x \leqslant 2\pi,$$ $$g : x \mapsto \frac{1}{2}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$

1. Solve the equation \(\text{fg}(x) = 1\). [3]
2. Sketch the graph of \(y = \text{f}(x)\). [3]

The function f is defined by \(\text{f} : x \mapsto 2x^2 - 12x + 7\) for \(x \in \mathbb{R}\).

1. Express \(2x^2 - 12x + 7\) in the form \(2(x + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
2. State the range of f. [1]

The function g is defined by \(\text{g} : x \mapsto 2x^2 - 12x + 7\) for \(x \leqslant k\).

1. State the largest value of \(k\) for which g has an inverse. [1]
2. Given that g has an inverse, find an expression for \(\text{g}^{-1}(x)\). [3]

The function f is defined by \(\mathrm{f} : x \mapsto 7 - 2x^2 - 12x\) for \(x \in \mathbb{R}\).

1. Express \(7 - 2x^2 - 12x\) in the form \(a - 2(x + b)^2\), where \(a\) and \(b\) are constants. [2]
2. State the coordinates of the stationary point on the curve \(y = \mathrm{f}(x)\). [1]

The function g is defined by \(\mathrm{g} : x \mapsto 7 - 2x^2 - 12x\) for \(x \geqslant k\).

1. State the smallest value of \(k\) for which g has an inverse. [1]
2. For this value of \(k\), find \(\mathrm{g}^{-1}(x)\). [3]