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### Section 1-1 : Functions

For problems 1 – 4 the given functions perform the indicated function evaluations.

1. $$f\left( x \right) = 3 - 5x - 2{x^2}$$ Solution
1. $$f\left( 4 \right)$$
2. $$f\left( 0 \right)$$
3. $$f\left( { - 3} \right)$$
1. $$f\left( {6 - t} \right)$$
2. $$f\left( {7 - 4x} \right)$$
3. $$f\left( {x + h} \right)$$
2. $$\displaystyle g\left( t \right) = \frac{t}{{2t + 6}}$$ Solution
1. $$g\left( 0 \right)$$
2. $$g\left( { - 3} \right)$$
3. $$g\left( {10} \right)$$
1. $$g\left( {{x^2}} \right)$$
2. $$g\left( {t + h} \right)$$
3. $$g\left( {{t^2} - 3t + 1} \right)$$
3. $$h\left( z \right) = \sqrt {1 - {z^2}}$$ Solution
1. $$h\left( 0 \right)$$
2. $$h\left( { - \frac{1}{2}} \right)$$
3. $$h\left( {\frac{1}{2}} \right)$$
1. $$h\left( {9z} \right)$$
2. $$h\left( {{z^2} - 2z} \right)$$
3. $$h\left( {z + k} \right)$$
4. $$\displaystyle R\left( x \right) = \sqrt {3 + x} - \frac{4}{{x + 1}}$$ Solution
1. $$R\left( 0 \right)$$
2. $$R\left( 6 \right)$$
3. $$R\left( { - 9} \right)$$
1. $$R\left( {x + 1} \right)$$
2. $$R\left( {{x^4} - 3} \right)$$
3. $$R\left( {\frac{1}{x} - 1} \right)$$

The difference quotient of a function $$f\left( x \right)$$ is defined to be,

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

For problems 5 – 9 compute the difference quotient of the given function.

1. $$f\left( x \right) = 4x - 9$$ Solution
2. $$g\left( x \right) = 6 - {x^2}$$ Solution
3. $$f\left( t \right) = 2{t^2} - 3t + 9$$ Solution
4. $$\displaystyle y\left( z \right) = \frac{1}{{z + 2}}$$ Solution
5. $$\displaystyle A\left( t \right) = \frac{{2t}}{{3 - t}}$$ Solution

For problems 10 – 17 determine all the roots of the given function.

1. $$f\left( x \right) = {x^5} - 4{x^4} - 32{x^3}$$ Solution
2. $$R\left( y \right) = 12{y^2} + 11y - 5$$ Solution
3. $$h\left( t \right) = 18 - 3t - 2{t^2}$$ Solution
4. $$g\left( x \right) = {x^3} + 7{x^2} - x$$ Solution
5. $$W\left( x \right) = {x^4} + 6{x^2} - 27$$ Solution
6. $$f\left( t \right) = {t^{\frac{5}{3}}} - 7{t^{\frac{4}{3}}} - 8t$$ Solution
7. $$\displaystyle h\left( z \right) = \frac{z}{{z - 5}} - \frac{4}{{z - 8}}$$ Solution
8. $$\displaystyle g\left( w \right) = \frac{{2w}}{{w + 1}} + \frac{{w - 4}}{{2w - 3}}$$ Solution

For problems 18 – 22 find the domain and range of the given function.

1. $$Y\left( t \right) = 3{t^2} - 2t + 1$$ Solution
2. $$g\left( z \right) = - {z^2} - 4z + 7$$ Solution
3. $$f\left( z \right) = 2 + \sqrt {{z^2} + 1}$$ Solution
4. $$h\left( y \right) = - 3\sqrt {14 + 3y}$$ Solution
5. $$M\left( x \right) = 5 - \left| {x + 8} \right|$$ Solution

For problems 23 – 32 find the domain of the given function.

1. $$\displaystyle f\left( w \right) = \frac{{{w^3} - 3w + 1}}{{12w - 7}}$$ Solution
2. $$\displaystyle R\left( z \right) = \frac{5}{{{z^3} + 10{z^2} + 9z}}$$ Solution
3. $$\displaystyle g\left( t \right) = \frac{{6t - {t^3}}}{{7 - t - 4{t^2}}}$$ Solution
4. $$g\left( x \right) = \sqrt {25 - {x^2}}$$ Solution
5. $$h\left( x \right) = \sqrt {{x^4} - {x^3} - 20{x^2}}$$ Solution
6. $$\displaystyle P\left( t \right) = \frac{{5t + 1}}{{\sqrt {{t^3} - {t^2} - 8t} }}$$ Solution
7. $$f\left( z \right) = \sqrt {z - 1} + \sqrt {z + 6}$$ Solution
8. $$\displaystyle h\left( y \right) = \sqrt {2y + 9} - \frac{1}{{\sqrt {2 - y} }}$$ Solution
9. $$\displaystyle A\left( x \right) = \frac{4}{{x - 9}} - \sqrt {{x^2} - 36}$$ Solution
10. $$Q\left( y \right) = \sqrt {{y^2} + 1} - \sqrt{{1 - y}}$$ Solution

For problems 33 – 36 compute $$\left( {f \circ g} \right)\left( x \right)$$ and $$\left( {g \circ f} \right)\left( x \right)$$ for each of the given pair of functions.

1. $$f\left( x \right) = 4x - 1$$, $$g\left( x \right) = \sqrt {6 + 7x}$$ Solution
2. $$f\left( x \right) = 5x + 2$$, $$g\left( x \right) = {x^2} - 14x$$ Solution
3. $$f\left( x \right) = {x^2} - 2x + 1$$, $$g\left( x \right) = 8 - 3{x^2}$$ Solution
4. $$f\left( x \right) = {x^2} + 3$$, $$g\left( x \right) = \sqrt {5 + {x^2}}$$ Solution