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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