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

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 1 } { x ^ { 2 } + 1 }\) for the domain \(0 \leqslant x \leqslant 2\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\), and hence that \(\mathrm { f } ( x )\) is an increasing function for \(x > 0\).
2. Find the range of \(\mathrm { f } ( x )\).
3. Given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 6 - 18 x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 3 } }\), find the maximum value of \(\mathrm { f } ^ { \prime } ( x )\). The function \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\).
4. Write down the domain and range of \(\mathrm { g } ( x )\). Add a sketch of the curve \(y = \mathrm { g } ( x )\) to a copy of Fig. 9 .
5. Show that \(\mathrm { g } ( x ) = \sqrt { \frac { x + 1 } { 2 - x } }\).

6 Given that \(\mathrm { f } ( x ) = \frac { x + 1 } { x - 1 }\), show that \(\mathrm { ff } ( x ) = x\).
Hence write down the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). What can you deduce about the symmetry of the curve \(y = \mathrm { f } ( x )\) ?

4 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h] \end{figure}

1. Find the value of \(a\).
2. Write down the range of \(\mathrm { f } ( x )\).
3. Sketch the curve \(y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1\).

7 You are given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are odd functions, defined for \(x \in \mathbb { R }\).

1. Given that \(\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\), prove that \(\mathrm { s } ( x )\) is an odd function.
2. Given that \(\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )\), determine whether \(\mathrm { p } ( x )\) is odd, even or neither.

3 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 1 - 2 \sin x\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Fig. 3 shows the curve \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure}

1. Write down the range of the function \(\mathrm { f } ( x )\).
2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\).
3. Find \(\mathrm { f } ^ { \prime } ( 0 )\). Hence write down the gradient of \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).

7

1. Show algebraically that the function \(\mathrm { f } ( x ) = \frac { 2 x } { 1 - x ^ { 2 } }\) is odd. Fig. 7 shows the curve \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 4\), together with the asymptote \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-4_730_817_431_607} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
2. Use the copy of Fig. 7 to complete the curve for \(- 4 \leqslant x \leqslant 4\).

7

1. The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \frac { 1 - x } { 1 + x } , x \neq - 1$$ Show that \(\mathrm { f } ( \mathrm { f } ( x ) ) = x\).
   Hence write down \(\mathrm { f } ^ { - 1 } ( x )\).
2. The function \(\mathrm { g } ( x )\) is defined for all real \(x\) by $$\mathrm { g } ( x ) = \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } }$$ Prove that \(\mathrm { g } ( x )\) is even. Interpret this result in terms of the graph of \(y = \mathrm { g } ( x )\).

3 The function f is defined by \(\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }\), for \(x \in \mathbb { R }\) and \(a > 0\).

1. For the curve with equation \(y = \mathrm { f } ( x )\),
   1. write down the equation of the asymptote,
   2. find the range of values that \(y\) can take.
   3. For the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\), write down
      (a) the equation of the line of symmetry,
      (b) the maximum and minimum values of \(y\),
   4. the set of values of \(x\) for which the curve is defined.

5

1. The graph of the function \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to \(P\) on each of the following curves.
   1. \(\quad y = \mathrm { f } ( x ) + 3\)
   2. \(\quad y = 2 \mathrm { f } ( 3 x - 1 )\)
   3. \(y = \mathrm { f } ^ { - 1 } ( x )\)
2. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-5_494_739_806_333} The diagram shows part of the graph of \(y = \mathrm { g } ^ { \prime } ( x )\). This is the graph of the gradient function of \(y = \mathrm { g } ( x )\). The graph intersects the \(x\)-axis at \(x = - 2\) and \(x = 4\).
   1. State the \(x\)-coordinate of any stationary points on the graph of \(y = \mathrm { g } ( x )\).
   2. State the set of values of \(x\) for which \(y = \mathrm { g } ( x )\) is a decreasing function.
   3. State the \(x\)-coordinate of any points of inflection on the graph of \(y = \mathrm { g } ( x )\).

8 Functions f and g are defined for \(0 \leqslant x \leqslant 2 \pi\) by \(\mathrm { f } ( x ) = 2 \tan x\) and \(\mathrm { g } ( x ) = \sec x\).

1. 1. State the range of f .
   2. State the range of \(g\).
2. 1. Show that \(\operatorname { fg } ( 0.6 ) = 5.33\), correct to 3 significant figures.
   2. Explain why \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( 0.6 )\) is not defined.
3. In this question you must show detailed reasoning. Solve the equation \(( \mathrm { f } ( x ) ) ^ { 2 } + 6 \mathrm {~g} ( x ) = 0\).

5

1. The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
   1. The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\). Find the values of \(a , b\) and \(c\).
   2. Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
2. The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
   1. Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
   2. State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.

5. The function f is defined by $$\mathrm { f } : x \rightarrow \frac { 2 x - 3 } { x - 1 } \quad x \in R , x \neq 1$$ a. Find \(f ^ { - 1 } ( 3 )\).
b. Show that $$\mathrm { ff } ( x ) = \frac { x + p } { x - 2 } \quad x \in R , \quad x \neq 2$$ where \(p\) is an integer to be found. The function g is defined by $$g : x \rightarrow x ^ { 2 } - 5 x \quad x \in R , 0 \leq x \leq 6$$ c. Find the range of g .
d. Explain why the function g does not have an inverse.

1. The figure 1 shows part of the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { a x + 4 } { x - b } , \quad x > 2\)

\begin{figure}[h] \end{figure} a. State the values of \(a\) and \(b\).
b. State the range of f.
c. Find \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.

10. The functions f and g are defined with their respective domains by $$\begin{array} { l l l } \mathrm { f } ( x ) = 4 - x ^ { 2 } & x \in R & x \geq 0 \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } & x \in R & x \geq 0 \end{array}$$ a. Write down the range of f .
b. Find the value of \(\mathrm { fg } ( 3 )\) c. Find \(\mathrm { g } ^ { - 1 } ( x )\)

5. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$

1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are integers to be found.
2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
3. 1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
   2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$

1. The function f is defined by

$$f ( x ) = 3 + \sqrt { x - 2 } \quad x \in \mathbb { R } \quad x > 2$$

1. State the range of f
2. Find f-1 The function \(g\) is defined by $$g ( x ) = \frac { 15 } { x - 3 } \quad x \in \mathbb { R } \quad x \neq 3$$
3. Find \(g f ( 6 )\)
4. Find the exact value of the constant \(a\) for which $$\mathrm { f } \left( a ^ { 2 } + 2 \right) = \mathrm { g } ( a )$$

1. The functions f and g are defined by

$$\begin{array} { l l } f ( x ) = 4 - 3 x ^ { 2 } & x \in \mathbb { R } \\ g ( x ) = \frac { 5 } { 2 x - 9 } & x \in \mathbb { R } , x \neq \frac { 9 } { 2 } \end{array}$$

1. Find fg(2)
2. Find \(\mathrm { g } ^ { - 1 }\)
3. 1. Find \(\mathrm { gf } ( x )\), giving your answer as a simplified fraction.
   2. Deduce the range of \(\operatorname { gf } ( x )\). The function h is defined by $$h ( x ) = 2 x ^ { 2 } - 6 x + k \quad x \in \mathbb { R }$$ where \(k\) is a constant.
4. Find the range of values of \(k\) for which the equation $$\mathrm { f } ( x ) = \mathrm { h } ( x )$$ has no real solutions.

1. The function f is defined by

$$f : x \mapsto \frac { 3 x - 5 } { x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - 1$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Show that $$\mathrm { ff } ( x ) = \frac { x + a } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq \pm 1$$ where \(a\) is an integer to be found. The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 3 x , \quad x \in \mathbb { R } , 0 \leqslant x \leqslant 5$$
3. Find the value of \(\mathrm { fg } ( 2 )\).
4. Find the range of g.
5. Explain why the function \(g\) does not have an inverse.

1. $$\operatorname { g } ( x ) = \frac { 2 x + 5 } { x - 3 } \quad x \geqslant 5$$

1. Find \(\mathrm { gg } ( 5 )\).
2. State the range of g.
3. Find \(\mathrm { g } ^ { - 1 } ( x )\), stating its domain.

6. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = \begin{cases} ( x - 2 ) ^ { 2 } + 1 & x \leqslant 2 \\ 4 x - 7 & x > 2 \end{cases}$$

1. Find the value of \(\operatorname { gg } ( 0 )\).
2. Find all values of \(x\) for which $$\mathrm { g } ( x ) > 28$$ The function h is defined by $$\mathrm { h } ( x ) = ( x - 2 ) ^ { 2 } + 1 \quad x \leqslant 2$$
3. Explain why h has an inverse but g does not.
4. Solve the equation $$\mathrm { h } ^ { - 1 } ( x ) = - \frac { 1 } { 2 }$$

1. The function f is defined by

$$f ( x ) = \frac { 8 x + 5 } { 2 x + 3 } \quad x > - \frac { 3 } { 2 }$$

1. Find \(\mathrm { f } ^ { - 1 } \left( \frac { 3 } { 2 } \right)\)
2. Show that $$\mathrm { f } ( x ) = A + \frac { B } { 2 x + 3 }$$ where \(A\) and \(B\) are constants to be found. The function \(g\) is defined by $$g ( x ) = 16 - x ^ { 2 } \quad 0 \leqslant x \leqslant 4$$
3. State the range of \(\mathrm { g } ^ { - 1 }\)
4. Find the range of \(\mathrm { fg } ^ { - 1 }\)

1. The functions f and g are defined by

$$\begin{aligned} & f ( x ) = 7 - 2 x ^ { 2 } \quad x \in \mathbb { R } \\ & \operatorname { g } ( x ) = \frac { 3 x } { 5 x - 1 } \quad x \in \mathbb { R } \quad x \neq \frac { 1 } { 5 } \end{aligned}$$

1. State the range of f
2. Find gf (1.8)
3. Find \(\mathrm { g } ^ { - 1 } ( x )\)

4. Given $$\begin{aligned} & \mathrm { f } ( x ) = \mathrm { e } ^ { x } , \quad x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 3 \ln x , \quad x > 0 , x \in \mathbb { R } \end{aligned}$$

1. find an expression for \(\mathrm { gf } ( x )\), simplifying your answer.
2. Show that there is only one real value of \(x\) for which \(\operatorname { gf } ( x ) = \operatorname { fg } ( x )\)

9 You are given that \(\mathrm { f } ( x ) = 2 x + 3 \quad\) for \(x < 0 \quad\) and \(\mathrm { g } ( x ) = x ^ { 2 } - 2 x + 1\) for \(x > 1\).

1. Find \(\mathrm { gf } ( x )\), stating the domain.
2. State the range of \(\mathrm { gf } ( x )\).
3. Find (gf) \({ } ^ { - 1 } ( x )\).

12 It is given that

• \(\mathrm { f } ( x ) = \pm \frac { 1 } { \sqrt { x } } , x > 0\)
• \(\mathrm { g } ( x ) = \frac { x } { x - 3 } , x > 3\)
• \(\mathrm { h } ( x ) = x ^ { 2 } + 2 , x \in \mathbb { R }\).
  1. Explain why \(\mathrm { f } ( x )\) is not a function.
  2. Find \(\mathrm { gh } ( x )\).
  3. State the domain of \(\mathrm { gh } ( x )\).