1.02l Modulus function: notation, relations, equations and inequalities

The graph of \(y = f(x)\) is shown below. \includegraphics{figure_1} One of the four equations listed below is the equation of the graph \(y = f(x)\) Identify which one is the correct equation of the graph. Tick (\(\checkmark\)) one box. [1 mark] \(y = |x + 2| + 3\) \(y = |x + 2| - 3\) \(y = |x - 2| + 3\) \(y = |x - 2| - 3\)

The function f is defined by $$f(x) = \left|\sin x + \frac{1}{2}\right| \quad (0 \leq x \leq 2\pi)$$ Find the set of values of \(x\) for which $$f(x) \geq \frac{1}{2}$$ Give your answer in set notation. [5 marks]

A curve has equation $$y = \frac{5 - 4x}{1 + x}$$

1. Sketch the curve. [4 marks]
2. Hence sketch the graph of \(y = \left|\frac{5 - 4x}{1 + x}\right|\). [1 mark]

The set \(A\) is defined by \(A = \{x : -\sqrt{2} < x < 0\} \cup \{x : 0 < x < \sqrt{2}\}\) Which of the inequalities given below has \(A\) as its solution? Circle your answer. [1 mark] \(|x^2 - 1| > 1\) \quad\quad \(|x^2 - 1| \geq 1\) \quad\quad \(|x^2 - 1| < 1\) \quad\quad \(|x^2 - 1| \leq 1\)

The diagram shows the line \(y = 5 - x\) \includegraphics{figure_15}

1. On the diagram above, sketch the graph of \(y = |x^2 - 4x|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) [3 marks]
2. Find the solution of the inequality $$|x^2 - 4x| > 5 - x$$ Give your answer in an exact form. [4 marks]

Solve the equation $$|2x + 1| = 3|x - 2|.$$ [4]

A function \(f\) is given by \(f(x) = |3x + 4|\).

1. Sketch the graph of \(y = f(x)\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis. [3]
2. On a separate set of axes, sketch the graph of \(y = \frac{1}{2}f(x) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A'\) and \(B'\). Clearly label the coordinates of the points \(A'\) and \(B'\). [3]

Solve the following inequalities giving your answer in set notation:

1. \(|4x + 3| < |x - 8|\) [3]
2. \(\frac{x}{x^2+1} < \frac{1}{2}\) [3]

\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$

1. State the range of \(f\) [1]
2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
3. state the set of possible values for \(k\). [2]

Solve the equation \(|2x - 1| = |x + 3|\). [3]

\includegraphics{figure_1} Figure 1 shows part of the graph of \(y = f(x)\), \(x \in \mathbb{R}\). The graph consists of two line segments that meet at the point \((1, a)\), \(a < 0\). One line meets the \(x\)-axis at \((3, 0)\). The other line meets the \(x\)-axis at \((-1, 0)\) and the \(y\)-axis at \((0, b)\), \(b < 0\). In separate diagrams, sketch the graph with equation

1. \(y = f(x + 1)\), [2]
2. \(y = f(|x|)\). [2]

Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(f(x) = |x - 1| - 2\), find

1. the value of \(a\) and the value of \(b\), [2]
2. the value of \(x\) for which \(f(x) = 5x\). [3]

\includegraphics{figure_4} Figure 4 Figure 4 shows a sketch of the graph with equation $$y = |2x - 3k|$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = f(x)\) where $$f(x) = k - |2x - 3k|$$ stating • the coordinates of the maximum point • the coordinates of any points where the graph cuts the coordinate axes [4]
2. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
3. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ [2]

You are given that \(gf(x) = |3x - 1|\) for \(x \in \mathbb{R}\).

1. Given that \(f(x) = 3x - 1\), express \(g(x)\) in terms of \(x\). [1]
2. State the range of \(gf(x)\). [1]
3. Solve the inequality \(|3x - 1| > 1\). [3]

Solve the equation \(3 + 2x = |7 - 4x|\). [3]

% 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]