1.02u Functions: definition and vocabulary (domain, range, mapping)

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]

\includegraphics{figure_9} The function f : \(x \mapsto p \sin^2 2x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \text{f}(x)\).

1. In terms of \(p\) and \(q\), state the range of f. [2]
2. State the number of solutions of the following equations.
   1. \(\text{f}(x) = p + q\) [1]
   2. \(\text{f}(x) = q\) [1]
   3. \(\text{f}(x) = \frac{1}{2}p + q\) [1]
3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\text{f}(x) = 4\), showing all necessary working. [5]

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]

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 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]

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

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

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

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

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

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 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]

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 $$f: x \mapsto |x - a| + a, \quad x \in \mathbb{R},$$ $$g: x \mapsto 4x + a, \quad x \in \mathbb{R},$$ 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]

\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]

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]

\includegraphics{figure_4} The function f is defined by \(f(x) = 2 - \sqrt{x}\) for \(x \geq 0\). The graph of \(y = f(x)\) is shown above.

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

\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]

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]

The functions f and g are defined for all real values of \(x\) by $$f(x) = 4x^2 - 12x \quad \text{and} \quad g(x) = ax + b,$$ where \(a\) and \(b\) are non-zero constants.

1. Find the range of f. [3]
2. Explain why the function f has no inverse. [2]
3. Given that \(g^{-1}(x) = g(x)\) for all values of \(x\), show that \(a = -1\). [4]
4. Given further that gf\((x) < 5\) for all values of \(x\), find the set of possible values of \(b\). [4]

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 function \(f(x) = \ln(1 + x^2)\) has domain \(-3 \leq x \leq 3\). Fig. 9 shows the graph of \(y = f(x)\). \includegraphics{figure_9}

1. Show algebraically that the function is even. State how this property relates to the shape of the curve. [3]
2. Find the gradient of the curve at the point P\((2, \ln 5)\). [4]
3. Explain why the function does not have an inverse for the domain \(-3 \leq x \leq 3\). [1]

The domain of \(f(x)\) is now restricted to \(0 \leq x \leq 3\). The inverse of \(f(x)\) is the function \(g(x)\).

1. Sketch the curves \(y = f(x)\) and \(y = g(x)\) on the same axes. State the domain of the function \(g(x)\). Show that \(g(x) = \sqrt{e^x - 1}\). [6]
2. Differentiate \(g(x)\). Hence verify that \(g'(\ln 5) = \frac{1}{4}\). Explain the connection between this result and your answer to part (ii). [5]

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]

The functions f(x) and g(x) are defined by $$f(x) = x^2, \quad g(x) = 2x - 1,$$ for all real values of \(x\).

1. State the ranges of f(x) and g(x). Explain why f(x) has no inverse. [3]
2. Find an expression for the inverse function g\(^{-1}\)(x) in terms of \(x\). Sketch the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\) on the same axes. [4]
3. Find expressions for gf(x) and fg(x). [2]
4. Solve the equation gf(x) = fg(x). Sketch the graphs of \(y = gf(x)\) and \(y = fg(x)\) on the same axes to illustrate your answer. [4]
5. Show that the equation f(x + a) = g\(^{-1}\)(x) has no solution if \(a > \frac{1}{4}\). [5]

The function f is defined by $$f : x \to 3e^{x-1}, \quad x \in \mathbb{R}.$$

1. State the range of f. [1]
2. Find an expression for \(f^{-1}(x)\) and state its domain. [4]

The function g is defined by $$g : x \to 5x - 2, \quad x \in \mathbb{R}.$$ Find, in terms of e,

1. the value of gf(ln 2), [3]
2. the solution of the equation $$f^{-1}g(x) = 4.$$ [4]

\(f(x) \equiv \frac{2x-3}{x-2}\), \(x \in \mathbb{R}\), \(x > 2\).

1. Find the range of \(f\). [2]
2. Show that \(f(f(x) = x\) for all \(x > 2\). [3]
3. Hence, write down an expression for \(f^{-1}(x)\). [1]

The function f is defined by $$\text{f}(x) \equiv 3 - x^2, \quad x \in \mathbb{R}, \quad x \geq 0.$$

1. State the range of f. [1]
2. Sketch the graphs of \(y = \text{f}(x)\) and \(y = \text{f}^{-1}(x)\) on the same diagram. [3]
3. Find an expression for f\(^{-1}(x)\) and state its domain. [3]

The function g is defined by $$\text{g}(x) \equiv \frac{8}{3-x}, \quad x \in \mathbb{R}, \quad x \neq 3.$$

1. Evaluate fg\((-3)\). [2]
2. Solve the equation $$\text{f}^{-1}(x) = \text{g}(x).$$ [3]