1.02k Simplify rational expressions: factorising, cancelling, algebraic division

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]

Using algebraic integration and making your method clear, find the exact value of $$\int_1^5 \frac{4x + 9}{x + 3} \, dx = a + \ln b$$ where \(a\) and \(b\) are constants to be found [4]

The function \(f\) is defined by $$f(x) = \frac{(x + 5)(x + 1)}{(x + 4)} - \ln(x + 4) \quad x \in \mathbb{R} \quad x > k$$

1. State the smallest possible value of \(k\). [1]
2. Show that $$f'(x) = \frac{ax^2 + bx + c}{(x + 4)^2}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4]
3. Hence show that \(f\) is an increasing function. [2]

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]

The curve \(C\) has equation $$y = \frac{x^2 - 2x - 3}{x + 2}.$$

1. Find the equations of the asymptotes of \(C\). [4]
2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]

% 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]

% Figure shows a curve with maximum at point A, passing through origin O, with horizontal asymptote \includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\) where $$f(x) = \frac{x^2 + 16}{3x} \quad x \neq 0$$ The curve has a maximum at the point \(A\) with coordinates \((a, b)\).

1. Find the value of \(a\) and the value of \(b\). [4] The function g is defined as $$g : x \mapsto \frac{x^2 + 16}{3x} \quad a \leq x < 0$$ where \(a\) is the value found in part (a).
2. Write down the range of g. [1]
3. On the same axes sketch \(y = g(x)\) and \(y = g^{-1}(x)\). [3]
4. Find an expression for \(g^{-1}(x)\) and state the domain of \(g^{-1}\) [5]
5. Solve the equation \(g(x) = g^{-1}(x)\). [3]