Functions

Domain and range

1. Find the domain of the following functions:
   

   (a)   $f(x)=3x-1$

   (b)   $f(x)=x^2+2x+3$

   (c)   $f(x)=\frac{x+1}{x}$

   Solutions:    (a)  domain $=\{x|x\in \mathbb{R}\}=\mathbb{R}$;     (b)  domain $=\{x|x\in \mathbb{R}\}=\mathbb{R}$;     (c)  domain $=\{x|x\in \mathbb{R},x\ne 0\}=\mathbb{R}\setminus \{0\}$
2. 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 $=\{x|x\in \mathbb{R},x⩾-2\}=[-2,\mathrm{\infty })$;     (b)  domain $=\{x|x\in \mathbb{R},x⩾\frac{3}{2}\}=[\frac{3}{2},\mathrm{\infty })$;     (c)  domain $=\{x|x\in \mathbb{R},x⩽3\}=(-\mathrm{\infty },3]$
3. 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 $=\{x\in \mathbb{R};x>-3\}=(-3,\mathrm{\infty })$;     (b)  domain $=\{x\in \mathbb{R};x<6\}=(-\mathrm{\infty },6)$;     (c)  domain $=\{x\in \mathbb{R};x<\frac{5}{2}\}=(-\mathrm{\infty },\frac{5}{2})$
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 $=\{x\in \mathbb{R};x\ne 0,x\ne 1\}$;     (b)  domain $=\{x\in \mathbb{R};x⩾-5\}$;     (c)  domain $=\{x\in \mathbb{R};x>0\}$
5. 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 $=\{x\in \mathbb{R};-2⩽x⩽2\}=[-2,2]$;     (b)  domain $=\{x\in \mathbb{R};-1<x⩽1\}=(-1,1]$;     (c)  domain $=\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$
6. 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 $=\{y|y\in \mathbb{R},y⩾-2\}=[-2,\mathrm{\infty })$;     (b)  range $=\{y|y\in \mathbb{R},0<y⩽1\}=(0,1]$;     (c)  range $=\{y|y\in \mathbb{R}\}=\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$
7. 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 $=(-2,\mathrm{\infty })$;     (b)  domain $=(-2,2)$, range $=\mathbb{R}$;     (c)  domain $=\{x;x\ne 2\}$, range $=\{y;y\ne 1\}$;     (d)  domain $=\mathbb{R}$, range $=\mathbb{R}$
8. Draw the graph of the function $f(x)=\frac{1}{x}$ for $x⩾1$ and then find the range of this function.
   Solutions:    range $=(0,1]$
9. Determine the range of the function $f:[-1,2]\to \mathbb{R},f(x)=x^2$.
   Solutions:    range $=[0,4]$
10. Determine the range of the function $f:\{x|x⩾1\}\to \mathbb{R},f:x\mapsto 2x-5$.
    Solutions:    range $=[-3,\mathrm{\infty })$
11. Determine the domain and range of the following functions.
    

    (a)        (b)

    Solutions:    (a)  domain $=\mathbb{R}$, range $=[-1,3]$;     (b)  domain $=(-\mathrm{\infty },-1)\cup (-1,1)\cup (1,\mathrm{\infty })$, range $=(-\mathrm{\infty },-1]\cup (0,\mathrm{\infty })$
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 $=[-1,1]$;     (b)  domain $=\mathbb{R}$, range $=[0,\mathrm{\infty })$;     (c)  domain $=(0,\mathrm{\infty })$, range $=[1,3)$
14. Determine the domain and range of the following functions.
    

    (a)        (b)

    Solutions:    (a)  domain $=[-5,1)$, range $=(-2,3]$;     (b)  domain $=(-\mathrm{\infty },2]$, range $=[-2,3]$

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