Functions

Polynomials

1. Find zeros (roots) of the following polynomials using factorisation:
   

   (a)   $p(x)=x^3+2x^2-15x$

   (b)   $p(x)=x^3-5x^2-4x+20$

   (c)   $p(x)=x^4-3x^3-x^2+3x$

   Solutions:    (a)  $x_1=0,x_2=3,x_3=-5$;     (b)  $x_1=2,x_2=-2,x_3=5$;     (c)  $x_1=0,x_2=1,x_3=-1,x_4=3$
2. Find zeros (roots) of the following polynomials using factorisation:
   

   (a)   $p(x)=x^3-2x^2-4x+8$

   (b)   $p(x)=x^4-4x^2$

   (c)   $p(x)=x^5-4x^4+4x^3$

   Solutions:    (a)  $x_{1,2}=2,x_3=-2$;     (b)  $x_{1,2}=0,x_3=2,x_4=-2$;     (c)  $x_{1,2,3}=0,x_{4,5}=2$
3. Use PolyRoots tool to find zeros of the following polynomials:
   

   (a)   $p(x)=x^3-x^2-14x+24$

   (b)   $p(x)=x^4-x^3-9x^2-11x-4$

   (c)   $p(x)=x^5-3x^4-5x^3+15x^2+4x-12$

   (d)   $p(x)=x^5-2x^4-5x^3+10x^2+4x-8$

   (e)   $p(x)=x^5-5x^4+4x^3-20x^2$

   Solutions:    (a)  $x_1=2,x_2=3,x_3=-4$;     (b)  $x_1=4,x_{2,3,4}=-1$;     (c)  $x_1=-1,x_2=1,x_3=-2,x_4=2,x_5=3$;     (d)  $x_{1,2}=2,x_3=-2,x_4=-1,x_5=1$;     (e)  $x_{1,2}=0,x_3=5$
4. Draw graphs and determine zeros of the following polynomials:
   

   (a)   $p(x)=x^3+x^2-2x$

   (b)   $p(x)=x^3-3x$

   (c)   $p(x)=x^4-5x^2+4$

   (d)   $p(x)=x^5-4x^3+3x$

   Solutions:    (a)  $x_1=0,x_2=1,x_3=-2$;     (b)  $x_1=0,x_2=\sqrt{3}\approx 1.73,x_3=-\sqrt{3}\approx -1.73$;     (c)  $x_1=1,x_2=-1,x_3=2,x_4=-2$;     (d)  $x_1=0,x_2=1,x_3=-1,x_4=\sqrt{3}\approx 1.73,x_5=-\sqrt{3}\approx -1.73$
5. Draw graphs and determine zeros of the following polynomials:
   

   (a)   $p(x)=x^4-5x^3+6x^2$

   (b)   $p(x)=x^4+x^3-2x^2$

   (c)   $p(x)=x^4-2x^3$

   (d)   $p(x)=x^5+4x^4+4x^3$

   Solutions:    (a)  $x_{1,2}=0,x_3=2,x_4=3$;     (b)  $x_{1,2}=0,x_3=1,x_4=-2$;     (c)  $x_{1,2,3}=0,x_4=2$;     (d)  $x_{1,2,3}=0,x_{4,5}=-2$
6. Use your GDC to draw graph of the polynomial $f(x)=2x^3-3x^2+5$
   

   (a)   Using GDC find zeros.

   (b)   Using GDC find $f(0),f(-3),f(\frac{1}{2})$ and $f(3.4)$.

   (c)   Using GDC find extreme points (maxima and minima).

   Solutions:    (a)  zero: $x_1=-1$;     (b)  $f(0)=5,f(-3)=-76,$ $f(\frac{1}{2})=\frac{9}{2}=4.5,f(3.4)\approx 48.9$;     (c)  maximum $P_1(0,5)$, minimum $P_2(1,4)$
7. Consider the polynomial $p(x)=x^3-2x^2+x-1$
   

   (a)   Draw the graph of this polynomial.

   (b)   Write down zeros.

   (c)   Calculate $p(0.02),p(\sqrt{3}),p(321)$.

   (d)   Find extreme points (maxima and minima).

   Solutions:    (b)  zero: $x_1\approx 1.75$;     (c)  $p(0.02)\approx -0.981,p(\sqrt{3})\approx -0.0718,p(321)\approx 3.29\cdot {10}^7$;     (d)  maximum $P_1(0.333,-0.852)$, minimum $P_2(1,-1)$
8. A polynomial has the equation $p(x)=x^3-3x^2+m$. Graph of this polynomial passes through the point $A(1,2)$.
   

   (a)   Find $m$.

   (b)   Draw the graph of this polynomial.

   (c)   Write down zeros.

   (d)   Find all values of $x$ where $p(x)=3$.

   (e)   Find extreme points (maxima and minima).

   Solutions:    (a)  $m=4$;     (c)  zeros: $x_1=-1,x_{2,3}=2$;     (d)  $x\approx -0.532,x\approx 0.653,x\approx 2.88$;     (e)  maximum $P_1(0,4)$, minimum $P_2(2,0)$
9. A polynomial has the equation $p(x)=x^3-6x^2+ax+b$. Graph of this polynomial passes through points $A(1,2)$. and $B(3,4)$.
   

   (a)   Find $a$ and $b$.

   (b)   Draw the graph of this polynomial.

   (c)   Write down zeros.

   This polynomial can be written as $p(x)=(x-2{)}^3+m$.

   (d)   Find $m$.

   Solutions:    (a)  $a=12,b=-5$;     (c)  zero: $x_1\approx 0.558$;     (d)  $m=3$
10. Graph of the function $y=x^3+a{x}^2+bx+c$ passes through points $A(-1,-5),B(1,3)$ and $C(2,-2)$.
    

    (a)   Find $a,b$ and $c$.

    (b)   Draw the graph of this function.

    (c)   Write the coordinates of extreme points.

    Solutions:    (a)  $a=-5,b=3,c=4$;     (c)  max.: $P_1(0.333,4.48)$, min.: $P_2(3,-5)$

Limits

11. ?
    ?

    Limit is the value that a function approaches as $x$ goes to positive or negative infinity.
    Evaluate the following limits:
    

    (a)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{2x+3}{x-1}}$

    (b)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x+2}{3x+5}}$

    (c)   ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{5+6x}{1-4x}}$

    Solutions:    (a)  $\cdots =2$;     (b)  $\cdots =\frac{1}{3}$;     (c)  $\cdots =-\frac{3}{2}$
12. Evaluate the following limits:
    

    (a)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2+x+1}{x^2+5x}}$

    (b)   ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{2x^2-x}{x^2+1}}$

    (c)   ${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}\frac{(x+1{)}^2}{2x(x+2)}}$

    (d)   ${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}\frac{3x^3+1}{(x+1{)}^3}}$

    Solutions:    (a)  $\cdots =1$;     (b)  $\cdots =2$;     (c)  $\cdots =\frac{1}{2}$;     (d)  $\cdots =3$
13. Evaluate the following limits (if possible):
    

    (a)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2+2x}{x^3+1}}$

    (b)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{1}{x^2+1}}$

    (c)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2}{2x+3}}$

    Solutions:    (a)  $\cdots =0$;     (b)  $\cdots =0$;     (c)  Not possible – the limit doesn't exist.

Rational functions

14. Write down zeros, vertical asymptotes and horizontal asymptotes of the following functions and draw the graphs:
    

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

    (b)   ${\displaystyle f(x)=\frac{2x-3}{x-1}}$

    (c)   ${\displaystyle f(x)=\frac{x+1}{2x+5}}$

    Solutions:    (a)  zero: $x=-1$, vertical asymptote: $x=1$, horizontal asymptote: $y=1$;     (b)  zero: $x=\frac{3}{2}$, vertical asymptote: $x=1$, horizontal asymptote: $y=2$;     (c)  zero: $x=-1$, vertical asymptote: $x=-\frac{5}{2}$, horizontal asymptote: $y=\frac{1}{2}$;
15. Write down zeros, vertical asymptotes and horizontal asymptotes of the following functions and draw the graphs:
    

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

    (b)   ${\displaystyle f(x)=\frac{2}{3x-4}}$

    (c)   ${\displaystyle f(x)=\frac{1}{1-x}}$

    Solutions:    (a)  zero: /, vertical asymptote: $x=-2$, horizontal asymptote: $y=0$;     (b)  zero: /, vertical asymptote: $x=\frac{4}{3}$, horizontal asymptote: $y=0$;     (c)  zero: /, vertical asymptote: $x=1$, horizontal asymptote: $y=0$;
16. Given the function ${\displaystyle f(x)=\frac{3x-3}{2x-1}}$
    

    (a)   write down the zero, vertical asymptote and horizontal asymptote,

    (b)   draw the graph,

    (c)   write the domain and range.

    Solutions:    (a)  zero: $x=1$, vertical asymptote: $x=\frac{1}{2}$, horizontal asymptote: $y=\frac{3}{2}$;     (c)  domain: $x\ne \frac{1}{2}$, range: $y\ne \frac{3}{2}$
17. Given the function ${\displaystyle f(x)=\frac{1-x}{2+x}}$
    

    (a)   write down the zero, vertical asymptote and horizontal asymptote,

    (b)   draw the graph,

    (c)   write the domain and range,

    (d)   find $f^{-1}(x)$.

    Solutions:    (a)  zero: $x=1$, vertical asymptote: $x=-2$, horizontal asymptote: $y=-1$;     (c)  domain: $x\ne -2$, range: $y\ne -1$;     (d)  $f^{-1}(x)=\frac{-2x+1}{x+1}$
18. A rational function has the equation ${\displaystyle f(x)=\frac{x^2}{x^2+x-2}}$.
    

    (a)   Draw the graph.

    (b)   Find zeros, vertical asymptotes and horizontal asymptote.

    (c)   Find the point where the graph intersects the horizontal asymptote.

    Solutions:    (b)  zero $x_{1,2}=0$, vertical asymptotes $x=1$ and $x=-2$, horizontal asymptote $y=1$;     (c)  intersection: $P(2,1)$
19. A rational function has the equation ${\displaystyle f(x)=\frac{x^2+4x+4}{x^2-1}}$.
    

    (a)   Draw the graph.

    (b)   Find zeros, vertical asymptotes and horizontal asymptote.

    (c)   Find the point where the graph intersects the horizontal asymptote.

    (d)   Find the extreme points.

    Solutions:    (b)  zero $x_{1,2}=-2$, vertical asymptotes $x=1$ and $x=-1$, horizontal asymptote $y=1$;     (c)  intersection: $P(-1.2,1)$;     (d)  min.: $(-2,0)$, max.: $(-0.5,-3)$
20. A rational function has the equation ${\displaystyle f(x)=\frac{x+1}{x^2}}$.
    

    (a)   Draw the graph.

    (b)   Find zeros, vertical asymptotes and horizontal asymptote.

    (c)   Find the extreme points.

    Solutions:    (b)  zero $x_1=-1$, vertical asymptotes $x_{1,2}=0$, horizontal asymptote $y=0$;     (c)  min.: $(-2,-0.25)$
21. A rational function has the equation ${\displaystyle f(x)=\frac{x-1}{x^2+5x+4}}$.
    

    (a)   Draw the graph.

    (b)   Find zeros, vertical asymptotes and horizontal asymptote.

    (c)   Find the extreme points. Round the coordinates to three decimals.

    Solutions:    (b)  zero $x_1=1$, vertical asymptotes $x=-1$ and $x=-4$, horizontal asymptote $y=0$;     (c)  min.: $(-2.162,1.481)$, max.: $(4.162,0.075)$
22. A rational function has the equation ${\displaystyle f(x)=\frac{x^2-3x}{x^2+1}}$.
    

    (a)   Draw the graph.

    (b)   Find zeros, vertical asymptotes and horizontal asymptote.

    (c)   Hence or otherwise, find the limit: ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2-3x}{x^2+1}}$

    Solutions:    (b)  zeros $x_1=0,x_2=3$, vertical asymptotes don't exist, horizontal asymptote $y=1$;     (c)  limit = 1
23. A rational function has the equation ${\displaystyle f(x)=\frac{x^3-2x^2}{x^2-2x+1}}$.
    

    (a)   Draw the graph.

    (b)   Find zeros, vertical asymptotes and horizontal asymptote.

    Solutions:    (b)  zeros $x_{1,2}=0,x_3=2$, vertical asymptote $x=1$, horizontal asymptote doesn't exist
24. A rational function has the equation ${\displaystyle f(x)=\frac{x}{x+1}-\frac{1}{x-1}}$.
    

    (a)   Draw the graph.

    (b)   Write down vertical asymptotes and horizontal asymptote.

    (c)   Write down zeros (using GDC).

    (d)   Find the zeros algebraically and write down the exact values.

    Solutions:    (b)  vertical asymptotes $x=1$ and $x=-1$, horizontal asymptote $y=1$;     (c)  zeros $x_1\approx -0.414,x_2\approx 2.41$;     (d)  zeros: $x_1=1-\sqrt{2},x_2=1+\sqrt{2}$
25. Let ${\displaystyle f(x)=\frac{2x}{x-q}}$.
    

    (a)   Write down the horizontal asymptote.

    The line $x=3$ is a vertical asymptote of this function.

    (b)   Find the value of $q$.

    (c)   Draw the graph of this function.

    Solutions:    (a)  horizontal asymptote $y=2$;     (b)  $q=3$
26. Let ${\displaystyle f(x)=\frac{x+2}{2x-q}}$. The line $x=2$ is a vertical asymptote of this function.
    

    (a)   Write down the horizontal asymptote.

    (b)   Find the value of $q$.

    (c)   Find the $y$-axis intercept of this function.

    Solutions:    (a)  horizontal asymptote $y=\frac{1}{2}$;     (b)  $q=4$;     (c)  $y=-\frac{1}{2}$

Modelling

27. We have a rectangular piece of cardboard with dimensions $80\times 60\mathrm{cm}$. We'd like to make a box out of this piece of cardboard. We'll cut off a small square at each corner, fold the sides and glue them together.
    

    (a)   Write the volume of this box as a function of $x$ (= the side of the small square).

    (b)   Draw this function in a coordinate system with appropriate units.

    (c)   Find the value of $x$ where the volume is maximal.

    (d)   Write down the maximal volume (in ${\mathrm{cm}}^3$ and in $\ell$).

    Solutions:    (a)  $V=x(80-2x)(60-2x)$;     (b)  (use $x$ from −10 to 50, $y$ from −5000 to 25000);     (c)  $x\approx 11.3\mathrm{cm}$;     (d)  $V\approx 24258{\mathrm{cm}}^3\approx 24.3\ell$
28. Doctors are studying the effects of a certain drug to the human body. They measured the concentration of this substance in blood and discovered the following model:
          ${\displaystyle y=\frac{85x}{x^2+5}}$
    where $y$ is concentration (in miligrams per litre) at time $x$ hours after taking a standard oral dose.
    

    (a)   Draw this function in a coordinate system with appropriate units.

    (b)   Find the value of $x$ where the concentration is maximal. Write $x$ in hours and minutes.

    (c)   Write down the maximal concentration (in $\mathrm{mg}/\ell$).

    (d)   When is the concentration equal to one half of the maximal value? Write the time in hours and minutes.

    Solutions:    (b)  $x_{\max }\approx 2\mathrm{h}14\min$;     (c)  $y_{\max }\approx 19.0\mathrm{mg}/\ell$;     (d)  $x_1\approx 0\mathrm{h}36\min$, $x_2\approx 8\mathrm{h}21\min$
29. A scientist was conducting a series of experiments. He was measuring two quantities, labelled as $x$ and $y$. The results of his measurements are written in the following table:
    $\begin{array}{|cc|}\hline x& y\\ -3& 2.5\\ -2& 1.7\\ -1& 1.2\\ 0& 1.0\\ 1& 1.2\\ 2& 1.7\\ 3& 2.5\\ \hline\end{array}$
    

    (a)   Draw the values as points in a coordinate system with appropriate units.

    (b)   Find an appropriate function $y=f(x)$ which can be used for modelling his results.

    Hint:    Use the function $y=x^2$ and apply transformations (stretch, shift) to adjust the graph.
    Solution:    (b)  $y=\frac{1}{6}{x}^2+1$
30. A scientist was conducting a series of experiments. She was measuring two quantities, labelled as $x$ and $y$. The results of her measurements are written in the following table:
    $\begin{array}{|cc|}\hline x& y\\ 0.0& 0.00\\ 0.5& 1.20\\ 1.0& 1.75\\ 1.5& 1.95\\ 2.0& 2.00\\ 2.5& 2.05\\ 3.0& 2.20\\ 3.5& 2.80\\ \hline\end{array}$
    

    (a)   Draw the values as points in a coordinate system with appropriate units.

    (b)   Find an appropriate function $y=f(x)$ which can be used for modelling her results.

    Hint:    Use the function $y=x^3$ and apply transformations (stretch, shift) to adjust the graph.
    Solution:    (b)  $y=\frac{1}{4}(x-2{)}^3+2$

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