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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 $$g\left( x \right) = \left\{ {\begin{array}{*{20}{l}}\frac{1}{2}x}&{{\mbox{if }}x \le 7}\\{{x^2} + 1}&{{\mbox{if }}7 < x < 11}\\{3 - x}&{{\mbox{if }}x \ge 11}\end{array}} \right$$ 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. $$\displaystyle g\left( t \right) = \sqrt {\frac{{6 + t}}{{5t - 10}}}$$