Functions

Transformations of graphs

1. Using your GDC draw the graph of the function \(y=x^2\). Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   \(y=x^2+1\)

   (b)   \(y=x^2-3\)

   (c)   \(y=(x-2)^2\)

   Solutions:    (a)  Translation of 1 unit parallel to the \(y\)-axis (shift upwards 1 unit);     (b)  translation of  −3 units parallel to the \(y\)-axis (shift downwards 3 units);     (c)  translation of 2 units parallel to the \(x\)-axis (shift right 2 units)
2. Using your GDC draw the graph of the function \(y=x^2\). Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   \(y=f(x)-2\)

   (b)   \(y=f(x-3)\)

   (c)   \(y=f(x+2)\)

   Solutions:    (a)  Translation of  −2 units along the \(y\)-axis (shift downwards 2 units);     (b)  translation of 3 units along the \(x\)-axis (shift right 3 units);     (c)  translation of  −2 units along the \(x\)-axis (shift left 2 units)
3. Using your GDC draw the graph of the function \(f(x)=\frac{\textstyle 1}{\textstyle x^2+1}\). Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   \(y=2\,f(x)\)

   (b)   \(y=3\,f(x)\)

   (c)   \(y=\frac{1}{2}\,f(x)\)

   Solutions:    (a)  Dilation along the \(y\)-axis by a scale factor 2 (stretch along \(y\)-axis by 2);     (b)  dilation along the \(y\)-axis by a scale factor 3 (stretch along \(y\)-axis by 3);     (c)  dilation along the \(y\)-axis by a scale factor \(\frac{1}{2}\) (stretch along \(y\)-axis by \(\frac{1}{2}\))
4. Using your GDC draw the graph of the function \(f(x)=\frac{\textstyle 1}{\textstyle x^2+1}\). Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   \(y=-\,f(x)\)

   (b)   \(y=-2\,f(x)\)

   (c)   \(y=-\frac{1}{2}\,f(x)\)

   Solutions:    (a)  Dilation along the \(y\)-axis by a scale factor  −1  =  reflection in the \(x\)-axis (flip upside down);     (b)  dilation along the \(y\)-axis by a scale factor  −2 (flip upside down and stretch along \(y\)-axis by 2);     (c)  dilation along the \(y\)-axis by a scale factor \(-\frac{1}{2}\) (flip upside down and stretch along \(y\)-axis by \(\frac{1}{2}\))
5. Draw the graph of the function \(f(x)=\log_2 x\). Then draw the following graphs manually  –  using transformations:
   

   (a)   \(y=2\,f(x)\)

   (b)   \(y=\frac{1}{2}\,f(x)\)

   (c)   \(y=-\,f(x)\)

   (d)   \(y=-2\,f(x)\)

   Solutions:    (a)  Dilation along the \(y\)-axis by a scale factor 2;     (b)  dilation along the \(y\)-axis by a scale factor \(\frac{1}{2}\);     (c)  reflection in the \(x\)-axis;     (d)  dilation along the \(y\)-axis by a scale factor  −2
6. Using your GDC draw the graph of the function \(f(x)=2^x\). Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   \(y=f(2x)\)

   (b)   \(y=f(3x)\)

   (c)   \(y=f(\frac{x}{2})\)

   Solutions:    (a)  Dilation along the \(x\)-axis by a scale factor \(\frac{1}{2}\) (stretch along \(x\)-axis by \(\frac{1}{2}\));     (b)  dilation along the \(x\)-axis by a scale factor \(\frac{1}{3}\) (stretch along \(x\)-axis by \(\frac{1}{3}\));     (c)  dilation along the \(x\)-axis by a scale factor 2 (stretch along \(x\)-axis by 2)
7. Using your GDC draw the graph of the function \(f(x)=2^x\). Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   \(y=f(-x)\)

   (b)   \(y=f(-2x)\)

   (c)   \(y=f(-\frac{x}{2})\)

   Solutions:    (a)  Dilation along the \(x\)-axis by a scale factor  −1  =  reflection in the \(y\)-axis (flip left-to-right);     (b)  dilation along the \(x\)-axis by a scale factor \(-\frac{1}{2}\) (flip left-to-right and stretch along \(x\)-axis by \(\frac{1}{2}\));     (c)  dilation along the \(x\)-axis by a scale factor  −2 (flip left-to-right and stretch along \(x\)-axis by 2)
8. Draw the graph of the function \(f(x)=\log_2 x\). Then draw the following graphs manually  –  using transformations:
   

   (a)   \(y=f(2x)\)

   (b)   \(y=f(-x)\)

   (c)   \(y=f(-2x)\)

   Solutions:    (a)  Dilation along the \(x\)-axis by a scale factor \(\frac{1}{2}\);     (b)  reflection in the \(y\)-axis;     (c)  dilation along the \(x\)-axis by a scale factor \(-\frac{1}{2}\)
9. The graph of the function \(y=f(x)\) is drawn in the coordinate system below. Draw the following graphs using transformations:
   

   (a)   \(y=f(x)-2\)

   (b)   \(y=f(x-2)\)

   (c)   \(y=2f(x)\)

   (d)   \(y=-\frac{1}{2}f(x)\)

   (e)   \(y=f\left(\frac{1}{2}x\right)\)

   

   Solutions:    (a)  Translation of  −2 units along \(y\)-axis;     (b)  translation of 2 units along \(x\)-axis;     (c)  dilation along the \(y\)-axis by a scale factor 2;     (d)  dilation along the \(y\)-axis by a scale factor \(-\frac{1}{2}\);     (e)  dilation along the \(x\)-axis by a scale factor 2
10. Use transformations to draw the graph of the function:   \(y=4-2^{x-2}\)
    Solutions:    (1)  Draw \(y=2^x\);     (2)  use reflection in \(x\)-axis (flip upside down) to obtain \(y=-2^x\);     (3)  use translation of 4 units along \(y\)-axis to obtain \(y=4-2^x\);     (4)  use translation of 2 units along \(x\)-axis to obtain \(y=4-2^{x-2}\)
11. Use transformations to draw the graph of the function:   \(y=\log_4 (2x+6)\)
    Solutions:    (1)  Draw \(y=\log_4 x\);     (2)  use dilation along the \(x\)-axis by a scale factor \(\frac{1}{2}\) to obtain \(y=\log_2 (2x)\);     (3)  use translation of  −3 units along \(x\)-axis to obtain \(y=\log_4 (2x+6)\)

Composite functions

12. Given the functions \(f(x)=3x+1\) and \(g(x)=x^2+2\) evaluate the following expressions:
    

    (a)   \(f(g(1))\)

    (b)   \((f\circ g)(-2)\)

    (c)   \((g\circ f)(-2)\)

    (d)   \((g\circ f)(5)\)

    Solutions:    (a)  \(f(g(1))=(f\circ g)(1)=10\);     (b)  \((f\circ g)(-2)=19\);     (c)  \((g\circ f)(-2)=27\);     (d)  \((g\circ f)(5)=258\)
13. Given the functions \(f(x)=3x+1\) and \(g(x)=x^2+2\) write down (and simplify) the composite functions:
    

    (a)   \((f\circ g)(x)\)

    (b)   \((g\circ f)(x)\)

    Solutions:    (a)  \((f\circ g)(x)=3(x^2+2)+1=3x^2+7\);     (b)  \((g\circ f)(x)=(3x+1)^2+2=9x^2+6x+3\)
14. Given the functions \(f(x)=\frac{\textstyle 1}{\textstyle x}\) and \(g(x)=5x-2\) write down the composite functions:
    

    (a)   \(f\circ g\)

    (b)   \(g\circ f\)

    Solutions:    (a)  \((f\circ g)(x)=\frac{1}{5x-2}\);     (b)  \((g\circ f)(x)=5\cdot\frac{1}{x}-2\)
15. Given the functions \(f\!:~x\mapsto\sqrt[\scriptstyle3]{x}\) and \(g\!:~x\mapsto \frac{\textstyle1}{\textstyle x+2}\) write down the composite functions:
    

    (a)   \(f\circ g\)

    (b)   \(g\circ f\)

    Solutions:    (a)  \((f\circ g)(x)=\sqrt[3]{\frac{1}{x+2}}\)   or   \(f\circ g\!:~x\mapsto\sqrt[3]{\frac{1}{x+2}}\);     (b)  \((g\circ f)(x)=\frac{1}{\sqrt[3]{x}+2}\)   or   \(g\circ f\!:~x\mapsto\frac{1}{\sqrt[3]{x}+2}\)
16. Given the functions \(f(x)=\frac{\textstyle x}{\textstyle x+1}\) and \(g(x)=\frac{\textstyle x+3}{\textstyle x}\) write down the composite functions:
    

    (a)   \((f\circ g)(x)\)

    (b)   \((g\circ f)(x)\)

    Solutions:    (a)  \((f\circ g)(x)=\frac{x+3}{2x+3}\);     (b)  \((g\circ f)(x)=\frac{4x+3}{x}\)
17. Given the functions \(f(x)=\frac{\textstyle x+1}{\textstyle x+5}\) and \(g(x)=\frac{\textstyle x+3}{\textstyle x+2}\) write down the composite functions:
    

    (a)   \(f(g(x))\)

    (b)   \(g(f(x))\)

    Solutions:    (a)  \((f\circ g)(x)=\frac{2x+5}{6x+13}\);     (b)  \((g\circ f)(x)=\frac{4x+16}{3x+11}\)
18. Given the functions \(f(x)=\frac{\textstyle x-2}{\textstyle x-3}\) and \(g(x)=\frac{\textstyle 3x-2}{\textstyle x-1}\) write down the composite functions:
    

    (a)   \(f\circ g\)

    (b)   \(g\circ f\)

    Solutions:    (a)  \((f\circ g)(x)=x\);     (b)  \((g\circ f)(x)=x\)
19. Given the function \(f(x)=\frac{\textstyle x+2}{\textstyle x+1}\) write down the composite function:
    

       \(f\circ f\)

    Solutions:    \((f\circ f)(x)=\frac{3x+4}{2x+3}\)
20. Given the functions \(f(x)=x^2+x\) and \(g(x)=x-3\):
    

    (a)   write down \((f\circ g)(x)\),

    (b)   solve the equation \((f\circ g)(x)=0\)

    Solutions:    (a)  \((f\circ g)(x)=x^2-5x+6\);     (b)  \(x_1=2,~ x_2=3\)
21. Given the functions \(f(x)=\frac{\textstyle x}{\textstyle x-2}\) and \(g(x)=\frac{\textstyle 3x+1}{\textstyle x-3}\):
    

    (a)   write down \(f(g(x))\)

    (b)   solve the equation \(f(g(x))=\frac{\textstyle1}{\textstyle2}\)

    Solutions:    (a)  \((f\circ g)(x)=\frac{3x+1}{x+7}\);     (b)  \(x=1\)
22. Given the functions \(f(x)=x^2-2x\) and \(g(x)=2x-1\) and \(h(x)=3x-4\):
    

    (a)   write down \(f\circ g\),

    (b)   solve the equation \((f\circ g)(x)=h(x)\)

    Solutions:    (a)  \((f\circ g)(x)=4x^2-8x+3\);     (b)  \(x_1=1,~ x_2=\frac{7}{4}\)
23. Given the functions \(f(x)=\frac{\textstyle x+2}{\textstyle x}\) and \(g(x)=\frac{\textstyle x+1}{\textstyle x+3}\) and \(h(x)=\frac{\textstyle x+5}{\textstyle x+1}\):
    

    (a)   write down \((f\circ g)(x)\),

    (b)   prove that \((h\circ h)(x)=\frac{\textstyle 3x+5}{\textstyle x+3}\),

    (c)   solve the equation \((f\circ g)(x)=(h\circ h)(x)\)

    Solutions:    (a)  \((f\circ g)(x)=\frac{3x+7}{x+1}\);     (c)  \(x=-2\)
24. Given the functions \(f(x)=x^2-2\) and \(g(x)=\frac{\textstyle x+1}{\textstyle 2}\) and \(k(x)=f(g(x))\):
    

    (a)   write down and simplify the equation of \(k(x)\),

    (b)   using GDC draw the graph of \(y=k(x)\),

    (c)   find the range of \(k(x)\)

    Solutions:    (a)  \(k(x)=\frac{1}{4}x^2+\frac{1}{2}x-\frac{7}{4}\);     (c)  range \(=[-2,\infty)\)

Inverse function

25. Find the inverse of each of the following functions:
    

    (a)   \(f(x)=2x+1\)

    (b)   \(f(x)=3x-2\)

    (c)   \(f(x)=\frac{1}{4}x+\frac{1}{2}\)

    Solutions:    (a)  \(f^{-1}(x)=\frac{x-1}{2}\);     (b)  \(f^{-1}(x)=\frac{x+2}{3}\);     (c)  \(f^{-1}(x)=4x-2\)
26. Find the inverse of each of the following functions:
    

    (a)   \(f(x)=x^3+2\)

    (b)   \(f(x)=2x^3\)

    (c)   \(f(x)=\frac{1}{4}x^3\)

    Solutions:    (a)  \(f^{-1}(x)=\sqrt[3]{x-2}\);     (b)  \(f^{-1}(x)=\sqrt[3]{\frac{x}{2}}\);     (c)  \(f^{-1}(x)=\sqrt[3]{4x}\)
27. Find the inverse of each of the following functions:
    

    (a)   \(f(x)=2^x\)

    (b)   \(f(x)=2^x+1\)

    (c)   \(f(x)=2^{x+1}\)

    Solutions:    (a)  \(f^{-1}(x)=\log_2 x\);     (b)  \(f^{-1}(x)=\log_2 (x-1)\);     (c)  \(f^{-1}(x)=\log_2 x-1\)
28. Given the function \(f(x)=\sqrt[\scriptstyle3]{x+3}\)
    

    (a)   write down the inverse function \(f^{-1}(x)\)

    (b)   draw graphs of \(f(x)\) and \(f^{-1}(x)\) in the same coordinate system.

    Solutions:    (a)  \(f^{-1}(x)=x^3-3\)
29. Find the inverse of each of the following functions:
    

    (a)   \({\displaystyle f(x)=\frac{x+5}{3x-2}}\)

    (b)   \({\displaystyle f(x)=\frac{3x}{x-2}}\)

    Solutions:    (a)  \(f^{-1}(x)=\frac{2x+5}{3x-1}\);     (b)  \(f^{-1}(x)=\frac{2x}{x-3}\)
30. Find the inverse of each of the following functions:
    

    (a)   \({\displaystyle f(x)=\frac{x-5}{x+2}}\)

    (b)   \({\displaystyle f(x)=\frac{1+x}{1-x}}\)

    Solutions:    (a)  \(f^{-1}(x)=\frac{2x+5}{-x+1}\);     (b)  \(f^{-1}(x)=\frac{x-1}{x+1}\)
31. Find the inverse of each of the following functions:
    

    (a)   \({\displaystyle f(x)=\frac{x+2}{x-1}}\)

    (b)   \({\displaystyle f(x)=\frac{2x}{x-2}}\)

    Solutions:    (a)  \(f^{-1}(x)=\frac{x+2}{x-1}\);     (b)  \(f^{-1}(x)=\frac{2x}{x-2}\)    —  in both cases \(f^{-1}(x)=f(x)\)
32. Prove that the function \({\displaystyle f(x)=\frac{3x+1}{x-3}}\) is its own inverse.
    Solutions:    You can prove it in two ways:    (a)  show that \(f^{-1}(x)=f(x)\)    or    (b)  show that \((f\circ f)(x)=x\)
33. Given the function \({\displaystyle f(x)=\frac{2x-5}{3x-4}}\) write down the following functions:
    

    (a)   \(f(x^{-1})\)

    (b)   \(\big(f(x)\big)^{-1}\)

    (c)   \(f^{-1}(x)\)

    Solutions:    (a)  \(f(x^{-1})=\frac{5x-2}{4x-3}\);     (b)  \(\big(f(x)\big)^{-1}=\frac{3x-4}{2x-5}\);     (c)  \(f^{-1}(x)=\frac{4x-5}{3x-2}\)
34. Given \({\displaystyle f(x)=\frac{x-5}{2}}\) and \({\displaystyle g(x)=\frac{x+3}{x}}\) write down the following functions:
    

    (a)   \(f^{-1}(x)\)

    (b)   \(g^{-1}(x)\)

    (c)   \((f\circ g)(x)\)

    (d)   \((f\circ g)^{-1}(x)\)

    (e)   \((g^{-1} \circ f^{-1})(x)\)

    Solutions:    (a)  \(f^{-1}(x)=2x+5\);     (b)  \(g^{-1}(x)=\frac{3}{x-1}\);     (c)  \((f\circ g)(x)=\frac{-4x+3}{2x}\);     (d,e)  \((f\circ g)^{-1}(x)=(g^{-1} \circ f^{-1})(x)=\frac{3}{2x+4}\)
35. Find the inverse of the function \(f(x)=x^2-3\)
    

    (a)   for \(x\geqslant 0\),

    (b)   for \(x\leqslant 0\).

    Solutions:    (a)  \(f^{-1}(x)=\sqrt{x+3}\);     (b)  \(f^{-1}(x)=-\sqrt{x+3}\)
36. Find the inverse of the function \(f(x)=\sqrt{x^2+4}\) for \(x\geqslant 0\). Write down the domain and range of \(f^{-1}\).
    Solutions:    \(f^{-1}(x)=\sqrt{x^2-4}\),   domain \(=[2,\infty)\),   range \(=[0,\infty)\)
37. Find the inverse of the function \(f(x)=\frac{\textstyle 4}{\textstyle x^2+1}\) for \(x\geqslant 0\). Write down the domain and range of \(f^{-1}\).
    Solutions:    \(f^{-1}(x)=\sqrt{\frac{4}{x}-1}\),   domain \(=(0,4]\),   range \(=[0,\infty)\)

 Index