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Section 3.4 : The Definition of a Function

For problems 1 – 6 determine if the given relation is a function.

1. \(\left\{ {\left( {0,1} \right),\left( {2,6} \right),\left( {9,4} \right),\left( {7,2} \right),\left( {12,3} \right)} \right\}\)
2. \(\left\{ {\left( { - 4,1} \right),\left( { - 2,1} \right),\left( {0,1} \right),\left( {3,1} \right)} \right\}\)
3. \(\left\{ {\left( {0,4} \right),\left( {0,6} \right),\left( {0,8} \right)} \right\}\)
4. \(\left\{ {\left( {1,6} \right),\left( { - 3,4} \right),\left( {7,6} \right),\left( {2, - 10} \right)} \right\}\)
5. \(\left\{ {\left( {0,1} \right),\left( {2,3} \right),\left( {4,5} \right),\left( {6,7} \right),\left( {8,9} \right),\left( {10,11} \right),\left( {12,13} \right)} \right\}\)
6. \(\left\{ {\left( { - 7,0} \right),\left( {4,2} \right),\left( {4,1} \right),\left( { - 2,3} \right),\left( {6,0} \right)} \right\}\)

For problems 7 – 13 determine if the given equation is a function.

1. \(\displaystyle y = \frac{2}{5}x + \frac{7}{5}\)
2. \(y = 3{x^2} + 4x + 1\)
3. \(y = 2 - {x^4}\)
4. \({y^2} = 10 - 3x\)
5. \({y^2} = {x^2} + 1\)
6. \({y^4} + {x^3} = 1\)
7. \({y^3} + {x^4} = 1\)
8. Given \(A\left( t \right) = 7t + 2\) determine each of the following.
   1. \(A\left( { - 9} \right)\)
   2. \(A\left( 0 \right)\)
   3. \(A\left( 2 \right)\)
   4. \(A\left( {6x} \right)\)
   5. \(A\left( {{t^2} + 1} \right)\)
9. Given \(f\left( x \right) = \frac{3}{x}\) determine each of the following.
   1. \(f\left( { - 4} \right)\)
   2. \(\displaystyle f\left( {\frac{1}{3}} \right)\)
   3. \(\displaystyle f\left( {\frac{6}{7}} \right)\)
   4. \(f\left( {4t + 2} \right)\)
   5. \(\displaystyle f\left( {\frac{6}{x}} \right)\)
10. Given \(h\left( w \right) = \sqrt {2w + 10} \) determine each of the following.
    1. \(h\left( { - 1} \right)\)
    2. \(h\left( 0 \right)\)
    3. \(h\left( 3 \right)\)
    4. \(h\left( { - 2t} \right)\)
    5. \(h\left( {w + 4} \right)\)
11. Given \(P\left( x \right) = 3 - 2x - {x^2}\) determine each of the following.
    1. \(P\left( { - 6} \right)\)
    2. \(P\left( 0 \right)\)
    3. \(P\left( 3 \right)\)
    4. \(P\left( {{z^2}} \right)\)
    5. \(P\left( {4 - x} \right)\)
12. Given \(f\left( z \right) = 2{z^3} - {z^2}\) determine each of the following.
    1. \(f\left( { - 1} \right)\)
    2. \(f\left( 0 \right)\)
    3. \(f\left( 4 \right)\)
    4. \(f\left( {\frac{1}{2}t} \right)\)
    5. \(f\left( {z - 1} \right)\)
13. Given \(g\left( t \right) = \left\{ {\begin{array}{*{20}{l}}{2 + t}&{{\mbox{if }}t \ge 10}\\{t - 7}&{{\mbox{if }}t < 10}\end{array}} \right.\) determine each of the following.
    1. \(g\left( {14} \right)\)
    2. \(g\left( {10} \right)\)
    3. \(g\left( { - 1} \right)\)
14. Given \(f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{4{x^2}}&{{\mbox{if }}x < - 4}\\{6x}&{{\mbox{if }}x \ge - 4}\end{array}} \right.\) determine each of the following.
    1. \(f\left( { - 6} \right)\)
    2. \(f\left( { - 4} \right)\)
    3. \(f\left( 3 \right)\)
15. Given \(g\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{\displaystyle \frac{1}{2}x}&{{\mbox{if }}x \le 7}\\{{x^2} + 1}&{{\mbox{if }}7 < x < 11}\\{3 - x}&{{\mbox{if }}x \ge 11}\end{array}} \right.\) determine each of the following.
    1. \(g\left( 2 \right)\)
    2. \(g\left( 7 \right)\)
    3. \(g\left( 8 \right)\)
    4. \(g\left( {11} \right)\)
    5. \(g\left( {14} \right)\)
16. Given \(A\left( w \right) = \left\{ {\begin{array}{*{20}{l}}{12}&{{\mbox{if }}w > - 8}\\{2 + 3w}&{{\mbox{if }} - 10 \le w \le - 8}\\{ - 1}&{{\mbox{if }}w < - 10}\end{array}} \right.\) determine each of the following.
    1. \(A\left( { - 12} \right)\)
    2. \(A\left( { - 10} \right)\)
    3. \(A\left( { - 9} \right)\)
    4. \(A\left( { - 8} \right)\)
    5. \(A\left( 0 \right)\)
17. Given \(f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{2x}&{{\mbox{if }}x < 6}\\{4 + x}&{{\mbox{if }}x = 6}\\{{x^2}}&{{\mbox{if }}x > 6}\end{array}} \right.\) determine each of the following.
    1. \(f\left( 0 \right)\)
    2. \(f\left( 2 \right)\)
    3. \(f\left( 6 \right)\)
    4. \(f\left( 8 \right)\)
    5. \(f\left( {10} \right)\)

For problems 24 – 28 compute the difference quotient for the given function. The difference quotient for the function \(f\left( x \right)\) is defined to be,

\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]

1. \(f\left( x \right) = 8x - 1\)
2. \(f\left( x \right) = 3{x^2}\)
3. \(f\left( x \right) = 7 - {x^2}\)
4. \(f\left( x \right) = 3{x^2} + 7x - 4\)
5. \(\displaystyle f\left( x \right) = \frac{2}{x}\)

For problems 29 – 39 determine the domain of the function.

1. \(f\left( x \right) = 9 - x\)
2. \(P\left( z \right) = {z^2} - 4\)
3. \(\displaystyle h\left( x \right) = \frac{{2 + x}}{{8x - 1}}\)
4. \(\displaystyle A\left( t \right) = \frac{{{t^2} - 4}}{{{t^2} + 6t - 7}}\)
5. \(\displaystyle h\left( w \right) = \frac{{{w^2} + 3w + 2}}{{{w^2} + 12w + 36}}\)
6. \(g\left( x \right) = \sqrt {10x - 15} \)
7. \(\displaystyle f\left( t \right) = \frac{{10t}}{{\sqrt {6 - 4t} }}\)
8. \(\displaystyle f\left( w \right) = \frac{{\sqrt {w + 7} }}{{\sqrt {2 - w} }}\)
9. \(A\left( z \right) = \sqrt {{z^2} - 9z} \)
10. \(h\left( z \right) = \sqrt {{z^2} - z - 20} \)
11. \(\displaystyle g\left( t \right) = \sqrt {\frac{{6 + t}}{{5t - 10}}} \)