Pauls Online Notes
Pauls Online Notes
Home / Algebra / Graphing and Functions / The Definition of a Function
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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