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

The one-one function f is defined by \(\mathrm{f}(x) = (x - 2)^2 + 2\) for \(x \geqslant c\), where \(c\) is a constant.

1. State the smallest possible value of \(c\). [1]

In parts (ii) and (iii) the value of \(c\) is 4.

1. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of \(\mathrm{f}^{-1}\). [3]
2. Solve the equation \(\mathrm{f f}(x) = 51\), giving your answer in the form \(a + \sqrt{b}\). [5]

$$f(x) = \frac{2}{x-1} - \frac{6}{(x-1)(2x+1)}, \quad x > 1.$$

1. Prove that \(f(x) = \frac{4}{2x+1}\). [4]
2. Find the range of \(f\). [2]
3. Find \(f^{-1}(x)\). [3]
4. Find the range of \(f^{-1}(x)\). [1]

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

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]

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

1. Prove that \(f^{-1}(x) = f(x)\) for all \(x \in \mathbb{R}, x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(ff(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function \(g\), defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(fg(-2)\). [3]
2. Sketch the graph of the inverse function \(g^{-1}\) and state its domain. [3]

The function \(h\) is defined by \(h: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function \(h\) and state its range. [3]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = f(x), x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = f^{-1}(x)\), [2]
2. \(y = 3f(2x)\). [3]

Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that \(f\) is defined by $$f : x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

1. state
   1. the value of \(c\),
   2. the range of \(f\). [3]
2. Find the value of \(d\), giving your answer to 3 decimal places. [3]

The function \(g\) is defined by $$g : x \to \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find \(fg(x)\), giving your answer in its simplest form. [3]

The functions \(f\) and \(g\) are defined by $$f: x \mapsto |x - a| + a, \quad x \in \mathbb{R},$$ $$g: x \mapsto 4x + a, \quad x \in \mathbb{R},$$ where \(a\) is a positive constant.

1. On the same diagram, sketch the graphs of \(f\) and \(g\), showing clearly the coordinates of any points at which your graphs meet the axes. [5]
2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of \(f\) and \(g\) intersect. [3]
3. Find an expression for \(fg(x)\). [2]
4. Solve, for \(x\) in terms of \(a\), the equation $$fg(x) = 3a.$$ [3]

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

1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
2. Find an expression for \(f^{-1}(x)\). [3]
3. Write down the domain of \(f^{-1}\). [1]

The functions f and g are defined with their respective domains by $$f(x) = x^2 \quad \text{for all real values of } x$$ $$g(x) = \frac{1}{2x + 1} \quad \text{for real values of } x, \quad x \neq -0.5$$

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

$$f(x) = \frac{2}{x - 1} - \frac{6}{(x - 1)(2x + 1)}, \quad x > 1$$

1. Prove that f(x) = \(\frac{4}{2x + 1}\). [4]
2. Find the range of f. [2]
3. Find \(f^{-1}(x)\). [3]
4. Find the range of \(f^{-1}(x)\). [1]

The function f is given by $$f: x \mapsto \ln(3x - 6), \quad x \in \mathbb{R}, \quad x > 2$$

1. Find \(f^{-1}(x)\). [3]
2. Write down the domain of \(f^{-1}\) and the range of \(f^{-1}\). [2]
3. Find, to 3 significant figures, the value of \(x\) for which f(x) = 3. [2]

The function g is given by $$g: x \mapsto \ln|3x - 6|, \quad x \in \mathbb{R}, \quad x \neq 2$$

1. Sketch the graph of \(y = g(x)\). [3]
2. Find the exact coordinates of all the points at which the graph of \(y = g(x)\) meets the coordinate axes. [3]

The function f is given by $$f : x \alpha \frac{x}{x^2-1} - \frac{1}{x+1}, \quad x > 1.$$

1. Show that \(\text{f}(x) = \frac{1}{(x-1)(x+1)}\). [3]
2. Find the range of f. [2]

The function g is given by $$g : x \alpha \frac{2}{x}, \quad x > 0.$$

1. Solve gf(x) = 70. [4]

The functions f and g are defined by $$\text{f}: x \alpha |x - a| + a, \quad x \in \mathbb{R},$$ $$\text{g}: x \alpha 4x + a, \quad x \in \mathbb{R}.$$ where \(a\) is a positive constant.

1. On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of any points at which your graphs meet the axes. [5]
2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect. [3]
3. Find an expression for fg(x). [2]
4. Solve, for \(x\) in terms of \(a\), the equation $$\text{fg}(x) = 3a.$$ [3]

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

1. Prove that \(\text{f}^{-1}(x) = \text{f}(x)\) for all \(x \in \mathbb{R}\), \(x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(\text{f}f(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function g, defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(\text{f}g(-2)\). [3]
2. Sketch the graph of the inverse function \(\text{g}^{-1}\) and state its domain. [3]

The function h is defined by \(\text{h}: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function h and state its range. [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]

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

1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
2. Find an expression for \(f^{-1}(x)\). [3]
3. Write down the domain of \(f^{-1}\). [1]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = \text{f}(x)\), \(x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = \text{f}^{-1}(x)\), [2]
2. \(y = 3\text{f}(2x)\). [3]

Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\text{f}: x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

1. state
   1. the value of \(c\),
   2. the range of f.
   [3]
2. Find the value of \(d\), giving your answer to 3 decimal places. [3]

The function g is defined by $$\text{g}: x \mapsto \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find fg(x), giving your answer in its simplest form. [3]

\includegraphics{figure_9} The function f is defined by \(f(x) = \sqrt{(mx + 7)} - 4\), where \(x \geq -\frac{7}{m}\) and \(m\) is a positive constant. The diagram shows the curve \(y = f(x)\).

1. A sequence of transformations maps the curve \(y = \sqrt{x}\) to the curve \(y = f(x)\). Give details of these transformations. [4]
2. Explain how you can tell that f is a one-one function and find an expression for \(f^{-1}(x)\). [4]
3. It is given that the curves \(y = f(x)\) and \(y = f^{-1}(x)\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5]

\includegraphics{figure_6} The diagram shows the graph of \(y = f(x)\), where $$f(x) = 2 - x^2, \quad x \leq 0.$$

1. Evaluate ff(-3). [3]
2. Find an expression for \(f^{-1}(x)\). [3]
3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]

Functions f and g are defined by $$f(x) = 2 \sin x \quad \text{for } -\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi,$$ $$g(x) = 4 - 2x^2 \quad \text{for } x \in \mathbb{R}.$$

1. State the range of f and the range of g. [2]
2. Show that gf(0.5) = 2.16, correct to 3 significant figures, and explain why fg(0.5) is not defined. [4]
3. Find the set of values of \(x\) for which \(f^{-1}g(x)\) is not defined. [6]

The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$

1. Evaluate ff(169). [2]
2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [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]

\includegraphics{figure_4} The function \(f\) is defined for all real values of \(x\) by $$f(x) = 2 - \sqrt{x + 1}.$$ The diagram shows the graph of \(y = f(x)\).

1. Evaluate \(f(-126)\). [2]
2. Find the set of values of \(x\) for which \(f(x) = |f(x)|\). [2]
3. Find an expression for \(f^{-1}(x)\). [3]
4. State how the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) are related geometrically. [1]

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 functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which

1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]