Review exercises  —  functions and calculator skills

1. Straight lines have the equations $y=2x-3$ and $y=-\frac{2}{3}x+5$.
   

   (a)   Draw these two straight lines.

   (b)   Find the intersection point.

   These two straight lines together with the axis of abscissas form a triangle $ABC$.

   (c)   Calculate the area of this triangle.

   (d)   Calculate the perimeter of this triangle.

   (e)   Is this a right-angled triangle? Justify your answer.

   Solutions:    (b)  $P(3,3)$;     (c)  $A=9$;     (d)  $P\approx 14.8$;     (e)  No, it isn't, because …
2. Consider the function $f(x)=x+|2x-3|$. The graph of $f$ and straight line $x-2y+9=0$ enclose a triangular region.
   

   (a)   Find all three vertices of this triangle.

   (b)   Find the area of this triangle.

   Solutions:    (a)  $A(1.5,1.5),B(-1,4),C(3,6)$;     (b)  $A=\frac{15}{2}=7.5$
3. Function $f$ has the equation:  $f(x)=2x^2-10x+8$.
   

   (a)   Draw the graph of this function.

   (b)   Write the function in the form $f(x)=a(x-h)(x-k)$.

   (c)   Find the vertex of $f$.

   Draw the function $g(x)=2x-8$ in the same coordinate system.

   (d)   Find the coordinates of the intersection points.

   (e)   Calculate the area enclosed by $f$ and $g$.

   Solutions:    (b)  $f(x)=2(x-1)(x-4)$;     (c)  $V(2.5,-4.5)$;     (d)  $P_1(2,-4),P_2(4,0)$;     (e)  $A=\frac{8}{3}\approx 2.67$
4. Function $f$ is a quadratic function. It intercepts the $x$-axis at 1 and 4. The $y$-axis intercept has coordinates $P(0,-4)$.
   

   (a)   Write the equation of $f$ and draw the graph.

   Graph of $f$ passes through points $A(2,m)$ and $B(5,n)$.

   (b)   Find $m$ and $n$.

   Straight line $L$ pases through $A$ and $B$, too.

   (c)   Write the equation of $L$.

   (d)   Calculate the area enclosed by $f$ and $L$.

   Solutions:    (a)  $f(x)=-x^2+5x-4$;     (b)  $m=2,n=-4$;     (c)  $y=-2x+6$;     (d)  $A=\frac{9}{2}=4.5$
5. Function $f$ has the equation $f(x)=x^3+x^2-x+2$.
   

   (a)   Draw the graph of $f$.

   (b)   Find $x$- and $y$-axis intercepts.

   (c)   Find extremes of $f$. Write the exact coordinates of extremes.

   Graph of $f$ has a tangent $t$ at point $P(1,f(1))$.

   (d)   Write the equation of $t$.

   Function $f$ and line $t$ have another point in common, besides $P$. We'll label this point $R$.

   (e)   Write the coordinates of $R$.

   Solutions:    (b)  $(-2,0)$ and $(0,2)$;     (c)  max: $(-1,3)$, min: $(\frac{1}{3},\frac{49}{27})$;     (d)  tangent: $y=4x-1$;     (e)  $R(-3,-13)$
6. Graph of the function $f(x)=x^3+a{x}^2+bx+c$ passes through points $A(1,7)$, $B(2,4)$ and $C(3,3)$.
   

   (a)   Find $a$, $b$ and $c$.

   (b)   Draw the graph of $f$.

   (c)   Write the equation of the tangent to $f$ at point $C$.

   (d)   Find the area of the region between the graph of $f$ and this tangent.

   Solutions:    (a)  $a=-5,b=5,c=6$;     (c)  $y=2x-3$;     (d)  $A=\frac{64}{3}=21\frac{1}{3}\approx 21.3$
7. Consider the functions $f(x)=\sqrt{2x+12}$  and  $g(x)=x^2$.
   

   (a)   Find the domain and range of the function $f$.

   (b)   Draw graphs of $f$ and $g$. Shade the region enclosed by these two graphs.

   (c)   Write the coordinates of both intersection points.

   (d)   Find the area of the shaded region giving your answer rounded to three significant figures.

   Solutions:    (a)  Domain: $x⩾-6$, range: $y⩾0$;     (c)  $A(-1.71,2.93),B(2,4)$;     (d)  $A\approx 8.62$
8. Consider the function $f(x)={\displaystyle \frac{x-4}{x-3}}$.
   

   (a)   Draw the graph of $f$ and write down the equations of its asymptotes.

   Line $L_1$ is the tangent to $f$ at $x=1$ and line $L_2$ is the tangent to $f$ at $x=4$ .

   (b)   Write equations of $L_1$ and $L_2$.

   (c)   Find the intersection point of $L_1$ and $L_2$.

   (d)   Find the area of the triangle formed by $L_2$ and both coordinate axes.

   Solutions:    (a)  asymptotes: $x=3$ and $y=1$;     (b)  $L_1:y=\frac{1}{4}x+\frac{5}{4}$,   $L_2:y=x-4$;     (c)  $P(7,3)$;     (d)  $A=8$
9. Function $f$ has the equation $f(x)={\displaystyle \frac{x+3}{x+1}}$.
   

   (a)   Draw the graph of $f$ and write down the domain and range.

   Function $f$ can be written in the form: $f(x)={\displaystyle p+\frac{q}{x+1}}$.

   (b)   Find $p$ and $q$.

   (c)   Calculate the integral ${\displaystyle \int f(x)dx}$.

   (d)   Hence or otherwise, calculate the area between the graf of $f$ and $x$-axis on the interval $0⩽x⩽3$.

   Solutions:    (a)  Domain: $x\ne -1$, range: $y\ne 1$;     (b)  $p=1,q=2$;     (c)  $\cdots =x+2\ln |x+1|+C$;     (d)  $A=3+\ln 16\approx 5.77$
10. Function $f$ has the equation $f(x)={\displaystyle \frac{x-1}{x-2}}$.
    

    (a)   Write the equation of the inverse function $f^{-1}$.

    Function $h$ is given as $h=f\circ f$.

    (b)   Show that $h$ can be written as $h(x)={\displaystyle \frac{1}{3-x}}$.

    (c)   Write the equation of the normal to $h$ at $x=1$.

    Solutions:    (a)  $f^{-1}(x)=\frac{2x-1}{x-1}$;     (c)  normal: $y=-4x+\frac{9}{2}$
11. Function $f$ has the equation $f(x)=\ln (x+e)$.
    

    (a)   Find the domain of $f$.

    (b)   Write the exact equation of the normal to $f$ at $x=0$.

    Function $g$ has the equation $g(x)=e^x$. Graphs of $f$ and $g$ have two intersection points: $A(0,1)$ and $B(p,q)$.

    (c)   Write $p$ and $q$ rounded to three significant figures.

    (d)   Calculate the area of the region enclosed by $f$ and $g$.

    Solutions:    (a)  Domain: $x>-e$;     (b)  normal: $y=-ex+1$;     (c)  $p\approx -1.46,q\approx 0.233$;     (d)  $A\approx 0.201$
12. Consider the function $f(x)=x\ln x-x$.
    

    (a)   Draw the graph of $f$.

    (b)   Find $f^{\prime }(x)$.

    (c)   Write the equation of the tangent to $f$ at $x=e$.

    (d)   Write the coordinates of the minimum.

    Solutions:    (b)  $f^{\prime }(x)=\ln x$;     (c)  tangent: $y=x-e$;     (d)  min.: $(1,-1)$
13. Consider the function $f(x)={\displaystyle \ln \frac{1+x}{1-x}}$.
    

    (a)   Find the domain of $f$.

    (b)   Write the equation of the tangent to $f$ at $x=0$.

    (c)   Find the equation of $f^{-1}$.

    (d)   Calculate the volume of the solid of revolution obtained by rotating the graph of $f^{-1}$ around $x$-axis for $0⩽x⩽2$.

    Solutions:    (a)  Domain: $-1<x<1$;     (b)  tangent: $y=2x$;     (c)  $f^{-1}(x)=\frac{e^x-1}{e^x+1}$;     (d)  $V\approx 1.50$
14. Consider the function $f(x)=\sin 2x$.
    

    (a)   Write $f^{\prime }(x)$.

    (b)   Write the equation of the tangent to $f$ at $x=2\pi$.

    Points $P$ and $Q$ are the $x$-axis intercepts for $-0.5⩽x⩽2.5$.

    (c)   Find the exact coordinates of $P$ and $Q$.

    (d)   Calculate the area of the region enclosed by $f$ and the $x$-axis between points $P$ and $Q$.

    Solutions:    (a)  $f^{\prime }(x)=2\cos 2x$;     (b)  tangent: $y=2x-4\pi$;     (c)  $P(0,0),Q(\frac{\pi }{2},0)$;     (d)  $A=1$
15. Consider the function $f(x)={\displaystyle \frac{\sin x-1}{\cos x}}$. Graph of this function passes through point $P(\pi ,q)$.
    

    (a)   Find $q$.

    (b)   Write $f^{\prime }(x)$.

    (c)   Write the equation of the normal to $f$ at $\pi$.

    Solutions:    (a)  $q=1$;     (b)  $f^{\prime }(x)=\frac{1-\sin x}{{\cos }^{2}x}$;     (c)  normal: $y=-x+\pi +1$

 Index