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)   \(f(x)=\frac{\textstyle x+1}{\textstyle x}\)

   Solutions:    (a)  domain \(=\{x|~ x\in\mathbb{R}\}=\mathbb{R}\);     (b)  domain \(=\{x|~ x\in\mathbb{R}\}=\mathbb{R}\);     (c)  domain \(=\{x|~ x\in\mathbb{R},~ x\ne0\}=\mathbb{R}\setminus\{0\}\)
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 \(=\{x|~ x\in\mathbb{R},~ x\geqslant -2\}=[-2,\infty)\);     (b)  domain \(=\left\{x|~ x\in\mathbb{R},~ x\geqslant \frac{3}{2}\right\}=\left[\frac{3}{2},\infty\right)\);     (c)  domain \(=\{x|~ x\in\mathbb{R},~ x\leqslant3\}=(-\infty,3]\)
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 \(=\{x\in\mathbb{R};~ x\gt -3\}=(-3,\infty)\);     (b)  domain \(=\{x\in\mathbb{R};~ x\lt6\}=(-\infty,6)\);     (c)  domain \(=\left\{x\in\mathbb{R};~ x\lt \frac{5}{2}\right\}=\left(-\infty,\frac{5}{2}\right)\)
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 \(=\{x\in\mathbb{R};~ x\ne0,~ x\ne1\}\);     (b)  domain \(=\{x\in\mathbb{R};~ x\geqslant -5\}\);     (c)  domain \(=\{x\in\mathbb{R};~ x\gt 0\}\)
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 \(=\{x\in\mathbb{R};~ -2\leqslant x\leqslant 2\}=[-2,2]\);     (b)  domain \(=\{x\in\mathbb{R};~ -1\lt x\leqslant 1\}=(-1,1]\);     (c)  domain \(=\mathbb{R}=(-\infty,\infty)\)
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 \(=\{y|~ y\in\mathbb{R},~ y\geqslant -2\}=[-2,\infty)\);     (b)  range \(=\{y|~ y\in\mathbb{R},~ 0\lt y\leqslant 1\}=(0,1]\);     (c)  range \(=\{y|~ y\in\mathbb{R}\}=\mathbb{R}=(-\infty,\infty)\)
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)   \(f(x)=\sqrt[\scriptstyle 3]{x}\)

   Solutions:    (a)  domain \(=\mathbb{R}\), range \(=(-2,\infty)\);     (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 \(f(x)=\frac{\textstyle 1}{\textstyle x}\) for \(x\geqslant 1\) and then find the range of this function.
   Solutions:    range \(=(0,1]\)
9. Determine the range of the function \(f\!:~ [-1,2]\rightarrow\mathbb{R},~~~ f(x)=x^2\).
   Solutions:    range \(=[0,4]\)
10. Determine the range of the function \(f\!:~ \{x|~ x\geqslant1\}\rightarrow\mathbb{R},~~~ f\!:~ x\mapsto 2x-5\).
    Solutions:    range \(=[-3,\infty)\)
11. Determine the domain and range of the following functions.
    

    (a)        (b)

    Solutions:    (a)  domain \(=\mathbb{R}\), range \(=[-1,3]\);     (b)  domain \(=(-\infty,-1)\cup(-1,1)\cup(1,\infty)\), range \(=(-\infty,-1]\cup(0,\infty)\)
12. Draw graphs of the following piecewise-defined functions:
    

    (a)   \(f(x)= \left\{\begin{array}{cl} x^2; & \mathrm{for}~ x\leqslant 1 \cr 2-x; & \mathrm{for}~ x\gt 1 \end{array} \right.\)

    (b)   \(f(x)= \left\{\begin{array}{cl} 2^x; & \mathrm{for}~ x\lt 1 \cr x; & \mathrm{for}~ x\geqslant 1 \end{array} \right.\)

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

    (a)   \(f(x)= \left\{\begin{array}{cl} -1; & \mathrm{for}~ x\lt -1 \cr x; & \mathrm{for}~ -1\leqslant x\leqslant 1 \cr 1; & \mathrm{for}~ x\gt 1 \end{array} \right.\)

    (b)   \(f(x)= \left\{\begin{array}{cl} -x; & \mathrm{for}~ x\leqslant 0 \cr \frac{\textstyle 1}{\textstyle x}; & \mathrm{for}~ x\gt 0 \end{array} \right.\)

    (c)   \(f(x)= \left\{\begin{array}{cl} x+1; & \mathrm{for}~ 0\lt x\lt 2 \cr 1; & \mathrm{for}~ x\geqslant 2 \end{array} \right.\)

    Solutions:    (a)  domain \(=\mathbb{R}\), range \(=[-1,1]\);     (b)  domain \(=\mathbb{R}\), range \(=[0,\infty)\);     (c)  domain \(=(0,\infty)\), 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 \(=(-\infty,2]\), range \(=[-2,3]\)

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