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Section 3.4 : The Definition of a Function

For problems 1 – 3 determine if the given relation is a function.

1. $\left\{ {\left( {2,4} \right),\left( {3, - 7} \right),\left( {6,10} \right)} \right\}$ Solution
2. $\left\{ {\left( { - 1,8} \right),\left( {4, - 7} \right),\left( { - 1,6} \right),\left( {0,0} \right)} \right\}$ Solution
3. $\left\{ {\left( {2,1} \right),\left( {9,10} \right),\left( { - 4,10} \right),\left( { - 8,1} \right)} \right\}$ Solution

For problems 4 – 6 determine if the given equation is a function.

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

For problems 12 & 13 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. $f\left( x \right) = 4 - 9x$ Solution
2. $f\left( x \right) = 2{x^2} - x$ Solution

For problems 14 – 18 determine the domain of the function.

1. $A\left( x \right) = 6x + 14$ Solution
2. $\displaystyle f\left( x \right) = \frac{1}{{{x^2} - 25}}$ Solution
3. $\displaystyle g\left( t \right) = \frac{{8t - 24}}{{{t^2} - 7t - 18}}$ Solution
4. $g\left( w \right) = \sqrt {9w + 7}$ Solution
5. $\displaystyle f\left( x \right) = \frac{1}{{\sqrt {{x^2} - 8x + 15} }}$ Solution