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

The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]

1. Give full details of the single transformation that transforms the graph of \(y = x^3\) to the graph of \(y = x^3 - 8\). [2]

The function f is defined by \(\mathrm{f}(x) = x^3 - 8\).

1. Find an expression for \(\mathrm{f}^{-1}(x)\). [2]
2. State how the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\) are related geometrically. [1]

The function f is defined by $$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$

1. Find the range of f. [1 mark]
2. 1. Find \(f^{-1}(x)\) [3 marks]
   2. State the range of \(f^{-1}(x)\) [1 mark]
3. State the transformation which maps the graph of \(y = f(x)\) onto the graph of \(y = f^{-1}(x)\) [1 mark]
4. Find the coordinates of the point of intersection of the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) [2 marks]

The function f is defined by \(f(x) = e^x + 1\) for \(x \in \mathbb{R}\) Find an expression for \(f^{-1}(x)\) Tick \((\checkmark)\) one box. [1 mark] \(f^{-1}(x) = \ln(x - 1)\) \(\square\) \(f^{-1}(x) = \ln(x) - 1\) \(\square\) \(f^{-1}(x) = \frac{1}{e^x + 1}\) \(\square\) \(f^{-1}(x) = \frac{x - 1}{e}\) \(\square\)

The function f is defined by $$f(x) = |x| + 1 \quad \text{for } x \in \mathbb{R}$$ The function g is defined by $$g(x) = \ln x$$ where g has its greatest possible domain.

1. Using set notation, state the range of f [2 marks]
2. State the domain of g [1 mark]
3. The composite function h is given by $$h(x) = g f(x) \quad \text{for } x \in \mathbb{R}$$
   1. Write down an expression for \(h(x)\) in terms of \(x\) [1 mark]
   2. Determine if h has an inverse. Fully justify your answer. [2 marks]

The function f is defined by $$f(x) = 4 + 3^{-x}, \quad x \in \mathbb{R}$$

1. Using set notation, state the range of f [2 marks]
2. The inverse of f is \(f^{-1}\)
   1. Using set notation, state the domain of \(f^{-1}\) [1 mark]
   2. Find an expression for \(f^{-1}(x)\) [3 marks]
3. The function g is defined by $$g(x) = 5 - \sqrt{x}, \quad (x \in \mathbb{R} : x > 0)$$
   1. Find an expression for \(gf(x)\) [1 mark]
   2. Solve the equation \(gf(x) = 2\), giving your answer in an exact form. [3 marks]

Each of these functions has domain \(x \in \mathbb{R}\) Which function does not have an inverse? Circle your answer. [1 mark] \(f(x) = x^3\) \quad \(f(x) = 2x + 1\) \quad \(f(x) = x^2\) \quad \(f(x) = e^x\)

The function f is defined by $$f(x) = 2x + 1$$ Solve the equation $$f(x) = f^{-1}(x)$$ Circle your answer. [1 mark] \(x = -1\) \quad\quad \(x = 0\) \quad\quad \(x = 1\) \quad\quad \(x = 2\)

The function f(x) is defined by \(\text{f}(x) = x^3 - 4\) for \(-1 \leq x \leq 2\). For \(\text{f}^{-1}(x)\), determine

• the domain
• the range.

[5]

Two real functions are defined as $$f(x) = \frac{8}{x-4} \quad \text{for} \quad (-\infty < x < 4) \cup (4 < x < \infty),$$ $$g(x) = (x-2)^2 \quad \text{for} \quad -\infty < x < \infty.$$

1. 1. Find an expression for \(fg(x)\). [2]
   2. Determine the values of \(x\) for which \(fg(x)\) does not exist. [3]
2. Find an expression for \(f^{-1}(x)\). [3]

The function \(f\) has domain \([4, \infty)\) and is defined by $$f(x) = \frac{2(3x + 1)}{x^2 - 2x - 3} + \frac{x}{x + 1}.$$

1. Show that \(f(x) = \frac{x + 2}{x - 3}\). [4]
2. Determine the range of \(f(x)\). [2]
3. Find an expression for \(f^{-1}(x)\) and write down the domain and range of \(f^{-1}\). [4]
4. Find the value of \(x\) when \(f(x) = f^{-1}(x)\). [4]

The functions \(f\) and \(g\) have domains \((-1, \infty)\) and \((0, \infty)\) respectively and are defined by $$f(x) = \cosh x, \qquad g(x) = x^2 - 1.$$

1. State the domain and range of \(fg\). [2]
2. Solve the equation \(fg(x) = 3\). Give your answer correct to three decimal places. [3]

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

1. Write down the domain and range of \(\text{f}^{-1}\) [2]
2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = \frac{5}{3}\) [3]

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

1. Write down the domain and range of \(\text{f}^{-1}\) [2]
2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = -\frac{5}{3}\) [3]

The function \(\mathbf{f}\) is defined by $$\mathbf{f}(x) = \frac{3x - 7}{x - 2} \quad x \in \mathbb{R}, x \neq 2$$

1. Find \(\mathbf{f}^{-1}(7)\) [2]
2. Show that \(\mathbf{f}(x) = \frac{ax + b}{x - 3}\) where \(a\) and \(b\) are integers to be found. [3]

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

1. Evaluate \(f(169)\). [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 diagram shows the graph of \(y = f(x)\), where \(f(x) = 2 - x^2, \quad x \leqslant 0\). \includegraphics{figure_8}

1. Evaluate \(f(-3)\). [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]

The function \(f\) is defined by $$f(x) = \frac{5x}{7x - 5}$$

1. The domain of \(f\) is the set \(\{x \in \mathbb{R} : x \neq a\}\) State the value of \(a\) [1 mark]
2. Prove that \(f\) is a self-inverse function [3 marks]
3. Find the range of \(f\) [1 mark]

The function \(f\) is defined by $$f(x) = \frac{12x}{3x + 4} \quad x \in \mathbb{R}, x \geq 0$$

1. Find the range of \(f\). [2]
2. Find \(f^{-1}\). [3]
3. Show, for \(x \in \mathbb{R}, x \geq 0\), that $$ff(x) = \frac{9x}{3x + 1}$$ [3]
4. Show that \(ff(x) = \frac{7}{2}\) has no solutions. [2]

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

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]

The functions f and g are defined for all real values of \(x\) by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]

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]

Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),

1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
2. Find the maximum value of \(f(x)\). [1]
3. Explain why \(f^{-1}(x)\) does not exist. [1]