Algebra - The Definition of a Function (Assignment Problems)

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Section 3.4 : The Definition of a Function

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

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

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

$\displaystyle y = \frac{2}{5}x + \frac{7}{5}$
$y = 3{x^2} + 4x + 1$
$y = 2 - {x^4}$
${y^2} = 10 - 3x$
${y^2} = {x^2} + 1$
${y^4} + {x^3} = 1$
${y^3} + {x^4} = 1$
Given A(t)=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. $A\left( {{t^2} + 1} \right)$
Given f(x)=3x 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. $\displaystyle f\left( {\frac{6}{x}} \right)$
Given h(w)=√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. $h\left( {w + 4} \right)$
Given P(x)=3−2x−x2 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. $P\left( {4 - x} \right)$
Given f(z)=2z3−z2 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. $f\left( {z - 1} \right)$
Given g(t)={2+tif t≥10t−7if t<10 determine each of the following.
1. $g\left( {14} \right)$
2. $g\left( {10} \right)$
3. $g\left( { - 1} \right)$
Given f(x)={4x2if x<−46xif x≥−4 determine each of the following.
1. $f\left( { - 6} \right)$
2. $f\left( { - 4} \right)$
3. $f\left( 3 \right)$
Given g(x)={12xif x≤7x2+1if 7<x<113−xif x≥11 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. $g\left( {14} \right)$
Given A(w)={12if w>−82+3wif −10≤w≤−8−1if w<−10 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. $A\left( 0 \right)$
Given f(x)={2xif x<64+xif x=6x2if x>6 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,

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

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

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

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