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 $f(x)=\frac{1}{x^2+1}$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   $y=2f(x)$

   (b)   $y=3f(x)$

   (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 $f(x)=\frac{1}{x^2+1}$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   $y=-f(x)$

   (b)   $y=-2f(x)$

   (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)   $y=2f(x)$

   (b)   $y=\frac{1}{2}f(x)$

   (c)   $y=-f(x)$

   (d)   $y=-2f(x)$

   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 $f(x)=\frac{1}{x}$ 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 $f:x\mapsto \sqrt[3]{x}$ and $g:x\mapsto \frac{1}{x+2}$ 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 $f(x)=\frac{x}{x+1}$ and $g(x)=\frac{x+3}{x}$ 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 $f(x)=\frac{x+1}{x+5}$ and $g(x)=\frac{x+3}{x+2}$ 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 $f(x)=\frac{x-2}{x-3}$ and $g(x)=\frac{3x-2}{x-1}$ 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 $f(x)=\frac{x+2}{x+1}$ 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 $f(x)=\frac{x}{x-2}$ and $g(x)=\frac{3x+1}{x-3}$:
    

    (a)   write down $f(g(x))$

    (b)   solve the equation $f(g(x))=\frac{1}{2}$

    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 $f(x)=\frac{x+2}{x}$ and $g(x)=\frac{x+1}{x+3}$ and $h(x)=\frac{x+5}{x+1}$:
    

    (a)   write down $(f\circ g)(x)$,

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

    (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 $g(x)=\frac{x+1}{2}$ 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 $=[-2,\mathrm{\infty })$

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 $f(x)=\sqrt[3]{x+3}$
    

    (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)   $(f(x){)}^{-1}$

    (c)   $f^{-1}(x)$

    Solutions:    (a)  $f(x^{-1})=\frac{5x-2}{4x-3}$;     (b)  $(f(x){)}^{-1}=\frac{3x-4}{2x-5}$;     (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 $x⩾0$,

    (b)   for $x⩽0$.

    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 $x⩾0$. Write down the domain and range of $f^{-1}$.
    Solutions:    $f^{-1}(x)=\sqrt{x^2-4}$,   domain $=[2,\mathrm{\infty })$,   range $=[0,\mathrm{\infty })$
37. Find the inverse of the function $f(x)=\frac{4}{x^2+1}$ for $x⩾0$. Write down the domain and range of $f^{-1}$.
    Solutions:    $f^{-1}(x)=\sqrt{\frac{4}{x}-1}$,   domain $=(0,4]$,   range $=[0,\mathrm{\infty })$

 Index