Functions

Domain and range

1. ?
   ?

   Domain of a function is the set of all $x$ values for which the function is defined.
   Find the domain of the following functions:
   

   (a)   $f(x)=\sqrt{x+2}$

   (b)   $f(x)=\sqrt{2x-3}$

   (c)   $f(x)=\sqrt{6-2x}$

   Solutions:    (a)  domain: $x⩾-2$;     (b)  domain: $x⩾\frac{3}{2}$;     (c)  domain: $x⩽3$
2. Find the domain of the following functions:
   

   (a)   $f(x)={\log }_2(x+3)$

   (b)   $f(x)={\log }_{\sqrt{5}}(6-x)$

   (c)   $f(x)=\ln (5-2x)$

   Solutions:    (a)  domain: $x>-3$;     (b)  domain: $x<6$;     (c)  domain: $x<\frac{5}{2}$
3. Use your GDC to draw the graph and then find the domain of the following functions:
   

   (a)   $f(x)=\sqrt{4-x^2}$

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

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

   Solutions:    (a)  domain: $-2⩽x⩽2$;     (b)  domain: $-1<x⩽1$;     (c)  domain: $\mathbb{R}$
4. Find the domain of the following functions:
   

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

   (b)   ${\displaystyle f(x)=\sqrt{\frac{x+5}{3}}}$

   (c)   $f(x)={\log }_{3}x+1$

   Solutions:    (a)  domain: $x\ne 0,x\ne 1$;     (b)  domain: $x⩾-5$;     (c)  domain: $x>0$
5. ?
   ?

   Range of a function is the set of all $y$ values which are the results of this function.
   Use your GDC to draw the graph and then find the range of the following functions:
   

   (a)   $f(x)=x^2-2$

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

   (c)   $f(x)=x^3-4x$

   Solutions:    (a)  range: $y⩾-2$;     (b)  range: $0<y⩽1$;     (c)  range: $\mathbb{R}$
6. Use your GDC to draw the graph and then find the domain and range of the following functions:
   

   (a)   $f(x)=e^x-2$

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

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

   (d)   $f(x)=\sqrt[3]{x}$

   Solutions:    (a)  domain: $\mathbb{R}$, range: $y>-2$;     (b)  domain: $-2<x<2$, range: $\mathbb{R}$;     (c)  domain: $x\ne 2$, range: $y\ne 1$;     (d)  domain: $\mathbb{R}$, range: $\mathbb{R}$
7. Draw the graph of the function $f(x)=\frac{1}{x}$, for $x⩾1$ and then find the range of this function.
   Solution:    range: $0<y⩽1$
8. Determine the range of the function $f(x)=x^2$, for $-1⩽x⩽2$.
   Solution:    range: $0⩽y⩽4$
9. Determine the range of the function $f(x)=2^x$, for $-2⩽x⩽3$.
   Solution:    range: $0.25⩽y⩽8$
10. Determine the range of the function $f(x)=2x-5$, for $x⩾1$.
    Solution:    range: $y⩾-3$
11. Determine the domain and range of the following functions.
    

    (a)        (b)

    Solutions:    (a)  domain: $\mathbb{R}$, range: $-1⩽y⩽3$;     (b)  domain: $x\ne -1,x\ne 1$, range: $y⩽-1,y>0$

Piecewise-defined functions

12. Draw graphs of the following piecewise-defined functions:
    

    (a)   $f(x)=\{\begin{array}{cl}{x}^2;& \mathrm{for}x⩽1\\ 2-x;& \mathrm{for}x>1\end{array}$

    (b)   $f(x)=\{\begin{array}{cl}{2}^x;& \mathrm{for}x<1\\ x;& \mathrm{for}x⩾1\end{array}$

    Solutions:
    
    (a)        (b)
13. Draw graphs and find the domain and range of the following functions:
    

    (a)   $f(x)=\{\begin{array}{cl}-1;& \mathrm{for}x<-1\\ x;& \mathrm{for}-1⩽x⩽1\\ 1;& \mathrm{for}x>1\end{array}$

    (b)   $f(x)=\{\begin{array}{cl}-x;& \mathrm{for}x⩽0\\ \frac{1}{x};& \mathrm{for}x>0\end{array}$

    (c)   $f(x)=\{\begin{array}{cl}x+1;& \mathrm{for}0<x<2\\ 1;& \mathrm{for}x⩾2\end{array}$

    Solutions:    (a)  domain: $\mathbb{R}$, range: $-1⩽y⩽1$;     (b)  domain: $\mathbb{R}$, range: $y⩾0$;     (c)  domain: $x>0$, range: $1⩽y<3$
14. Determine the domain and range of the following functions.
    

    (a)        (b)

    Solutions:    (a)  domain: $-5⩽x<1$, range: $-2<y⩽3$;     (b)  domain: $x⩽2$, range: $-2⩽y⩽3$

 Index