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Section 1.1 : Functions

For problems 1 – 6 the given functions perform the indicated function evaluations.

1. \(f\left( x \right) = 10x - 3\)
   1. \(f\left( { - 5} \right)\)
   2. \(f\left( 0 \right)\)
   3. \(f\left( 7 \right)\)
   1. \(f\left( {{t^2} + 2} \right)\)
   2. \(f\left( {12 - x} \right)\)
   3. \(f\left( {x + h} \right)\)
2. \(h\left( y \right) = 4{y^2} - 7y + 1\)
   1. \(h\left( 0 \right)\)
   2. \(h\left( { - 3} \right)\)
   3. \(h\left( 5 \right)\)
   1. \(h\left( {6z} \right)\)
   2. \(h\left( {1 - 3y} \right)\)
   3. \(h\left( {y + k} \right)\)
3. \(g\left( t \right) = \displaystyle \frac{{t + 5}}{{1 - t}}\)
   1. \(g\left( 0 \right)\)
   2. \(g\left( 4 \right)\)
   3. \(g\left( { - 7} \right)\)
   1. \(g\left( {{x^2} - 5} \right)\)
   2. \(g\left( {t + h} \right)\)
   3. \(g\left( {4\sqrt t + 9} \right)\)
4. \(f\left( z \right) = \sqrt {4z + 5} \)
   1. \(f\left( 0 \right)\)
   2. \(f\left( { - 1} \right)\)
   3. \(f\left( { - 2} \right)\)
   1. \(f\left( {5 - 12y} \right)\)
   2. \(f\left( {2{z^2} + 8} \right)\)
   3. \(f\left( {z + h} \right)\)
5. \(\displaystyle z\left( x \right) = \frac{{\sqrt {{x^2} + 9} }}{{4x + 8}}\)
   1. \(z\left( 4 \right)\)
   2. \(z\left( { - 4} \right)\)
   3. \(z\left( 1 \right)\)
   1. \(z\left( {2 - 7x} \right)\)
   2. \(z\left( {\sqrt {3x + 4} } \right)\)
   3. \(z\left( {x + h} \right)\)
6. \(\displaystyle Y\left( t \right) = \sqrt {3 - t} - \frac{t}{{2t + 5}}\)
   1. \(Y\left( 0 \right)\)
   2. \(Y\left( 7 \right)\)
   3. \(Y\left( { - 4} \right)\)
   1. \(Y\left( {5 - t} \right)\)
   2. \(Y\left( {{t^2} - 10} \right)\)
   3. \(Y\left( {6t - {t^2}} \right)\)

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 7 – 13 compute the difference quotient of the given function.

1. \(Q\left( t \right) = 4 - 7t\)
2. \(g\left( t \right) = 42\)
3. \(H\left( x \right) = 2{x^2} + 9\)
4. \(z\left( y \right) = 3 - 8y - {y^2}\)
5. \(g\left( z \right) = \sqrt {4 + 3z} \)
6. \(\displaystyle y\left( x \right) = \frac{{ - 4}}{{1 - 2x}}\)
7. \(\displaystyle f\left( t \right) = \frac{{{t^2}}}{{t + 7}}\)

For problems 14 – 21 determine all the roots of the given function.

1. \(y\left( t \right) = 40 + 3t - {t^2}\)
2. \(f\left( x \right) = 6{x^4} - 5{x^3} - 4{x^2}\)
3. \(Z\left( p \right) = 6 - 11p - {p^2}\)
4. \(h\left( y \right) = 4{y^6} + 10{y^5} + {y^4}\)
5. \(g\left( z \right) = {z^7} + 6{z^4} - 16z\)
6. \(f\left( t \right) = {t^{\frac{1}{2}}} - 8{t^{\frac{1}{4}}} + 15\)
7. \(\displaystyle h\left( w \right) = \frac{w}{{4w + 5}} + \frac{{3w}}{{w - 8}}\)
8. \(\displaystyle g\left( w \right) = \frac{w}{{w + 3}} - \frac{{w + 2}}{{4w - 1}}\)

For problems 22 – 30 find the domain and range of the given function.

1. \(f\left( x \right) = {x^2} - 8x + 3\)
2. \(z\left( w \right) = 4 - 7w - {w^2}\)
3. \(g\left( t \right) = 3{t^2} + 2t - 3\)
4. \(g\left( x \right) = 5 - \sqrt {2x} \)
5. \(B\left( z \right) = 10 + \sqrt {9 + 7{z^2}} \)
6. \(h\left( y \right) = 1 + \sqrt {6 - 7y} \)
7. \(f\left( x \right) = 12 - 5\sqrt {2x + 9} \)
8. \(V\left( t \right) = - 6\left| {5 - t} \right|\)
9. \(y\left( x \right) = 12 + 9\left| {{x^2} - 1} \right|\)

For problems 31 – 51 find the domain of the given function.

1. \(\displaystyle f\left( t \right) = \frac{{4 - 12t + 8{t^2}}}{{16t + 9}}\)
2. \(\displaystyle v\left( y \right) = \frac{{{y^3} - 27}}{{4 - 17y}}\)
3. \(\displaystyle g\left( x \right) = \frac{{3x + 1}}{{5{x^2} - 3x - 2}}\)
4. \(\displaystyle h\left( t \right) = \frac{{{t^3} - {t^2} + 1 - 1}}{{35{t^3} + 2{t^4} - {t^5}}}\)
5. \(\displaystyle f\left( z \right) = \frac{{{z^2} + z}}{{{z^3} - 9{z^2} + 2z}}\)
6. \(\displaystyle V\left( p \right) = \frac{{3 - {p^4}}}{{4{p^2} + 10p + 2}}\)
7. \(g\left( z \right) = \sqrt {{z^2} - 15} \)
8. \(f\left( t \right) = \sqrt {36 - 9{t^2}} \)
9. \(A\left( x \right) = \sqrt {15x - 2{x^2} - {x^3}} \)
10. \(Q\left( y \right) = \sqrt {4{y^3} - 4{y^2} + y} \)
11. \(\displaystyle P\left( t \right) = \frac{{{t^2} + 7}}{{\sqrt {6t - {t^2}} }}\)
12. \(\displaystyle h\left( t \right) = \frac{{{t^2}}}{{\sqrt {5 + 3t - {t^2}} }}\)
13. \(\displaystyle h\left( x \right) = \frac{6}{{\sqrt {{x^2} - 7x + 3} }}\)
14. \(\displaystyle f\left( z \right) = \frac{{z + 1}}{{\sqrt {{z^4} - 6{z^3} + 9{z^2}} }}\)
15. \(S\left( t \right) = \sqrt {8 - t} + \sqrt {2t} \)
16. \(g\left( x \right) = \sqrt {5x - 8} - 2\sqrt {x + 9} \)
17. \(h\left( y \right) = \sqrt {49 - {y^2}} - \frac{y}{{\sqrt {4y - 12} }}\)
18. \(\displaystyle A\left( x \right) = \frac{{x + 1}}{{x - 4}} + 4\sqrt {{x^2} + 10x + 9} \)
19. \(\displaystyle f\left( t \right) = \frac{8}{{{t^2} - 3t - 4}} + \frac{3}{{\sqrt {12 - 7t - 3{t^2}} }}\)
20. \(\displaystyle R\left( x \right) = \frac{3}{{{x^4} + {x^2}}} + \sqrt[5]{{{x^2} - x - 6}}\)
21. \(C\left( z \right) = {z^3} - \sqrt[4]{{{z^6} + {z^2}}}\)

For problems 52 – 55 compute \(\left( {f \circ g} \right)\left( x \right)\) and \(\left( {g \circ f} \right)\left( x \right)\) for each of the given pairs of functions.

1. \(f\left( x \right) = 5 + 2x\), \(g\left( x \right) = 8 - 23x\)
2. \(f\left( x \right) = \sqrt {2 - x} \), \(g\left( x \right) = 2{x^2} - 9\)
3. \(f\left( x \right) = 2{x^2} + x - 4\), \(g\left( x \right) = 7x - {x^2}\)
4. \(\displaystyle f\left( x \right) = \frac{x}{{3 + 2x}}\), \(g\left( x \right) = 8 + 5x\)